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prompt string | gt_answer string | final_answer string | initial_reason_steps string | A0 string | A1 string | A2 string | A3 string | score_A0 int64 | score_A1 float64 | score_A2 float64 | score_A3 float64 | mask_A0 int64 | mask_A1 int64 | mask_A2 int64 | mask_A3 int64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b. | ## Step 1: Ensuring continuity for $x > 2$ and $x > -2$
For the piecewise function to be continuous, we need the expressions for $x > 2$ and $x < -2$ to be equal to the expression for $2 \le x \le 2$ at $x = 2$ and $x = -2$, respectively. We have that $\lim_{x\to2^-} f(x) = \lim_{x\to2^+} f(x) = 2$ and $\lim_{x\to -2^-... | ## Step 1: Find the value of b at x = -2
Since f(x) is continuous, the values of f(-2) from the first two pieces must be equal. f(-2) from the second piece is -2 - 5 = -7. From the first piece, we use the limit as x approaches -2 to find f(-2), because the first piece is not directly defined at x = -2. We get f(-2) = a... | ## Step 1: The function needs to be continuous for x = 2
Since f(x) is continuous for x = 2, we can equate the two pieces of the function at x = 2:
\[2 - 5 = 2a + 3.\]
| 1 | 0.746758 | 0.662637 | 0.661406 | 1 | 1 | 1 | 1 | |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
| ## Step 2: First, let's consider the point where x = 2.
We need to equate ax + 3 and x - 5 at x = 2. So, we substitute 2 into both expressions to get a(2) + 3 = 2 - 5. | ## Step 2: To solve for a and b, we first consider the boundary point x = 2.
We know that f(2) should give us the same result, regardless of whether we use the expression ax+3 or x-5. So we have:
f(2) = 2a + 3 = 2 - 5
| ## Step 2: First, we'll set the two parts of the function equal to each other at x = 2.
The equations are ax + 3 = x - 5 at x = 2. Plugging in x = 2, we get:
2a + 3 = 2 - 5
2a = -6
| ## Step 2: To find the value of b, we use x = -2 in the second part of the piecewise function (since it is continuous for -2 ≤ x ≤ 2).
We get -2 - 5 = 2(-2) - b. This simplifies to -7 = -4 - b.
| 1 | 0.81502 | 0.805059 | 0.77 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 3: Form an equation by equating the two expressions for n
Since the total number of band members $n$ is the same in both formations, we can equate the two expressions and obtain $mr + 2 = (m + 1)(r - 2)$. | ## Step 3: Equate the two expressions for n and expand
Substituting the expression for $n$ from the initial formation into the equation for the new formation and expanding, we have $mr + 2 = m(r - 2) + 1$.
| ## Step 3: Equate the two expressions for n and simplify
Set the expressions for $n$ equal to each other: $mr + 2 = (m + 1)(r - 2)$. Expanding the right-hand side, we get $mr + 2 = mr - 2m + r - 2$, and combining like terms, we obtain $2m + 4 = r - 2m$, so $r = 4m + 6$.
| ## Step 3: Solve for mr
Equating the two expressions for $n$, we get $(m + 1)(r - 2) = mr + 2$. Expanding and simplifying, we have $mr - 2m = mr - 2m + 2 - 2$, which simplifies to $0 = 2 - 2$. However, equating the two expressions for $n$ gives $mr + 2 = mr - 2m + r - 2$.
| 1 | 0.806484 | 0.768496 | 0.767949 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 4: Expand and rearrange the equation
Expanding and rearranging the equation, we get $mr + 2 = mr - 2m + r - 2$. This simplifies to $2m + 2 = r - 2m$ and then to $4m + 2 = r$. | ## Step 4: Expand and simplify the equation from Step 3
Expanding and simplifying the equation $mr + 2 = (m + 1)(r - 2)$ yields $mr + 2 = mr - 2m + r - 2$, which simplifies to $mr - mr + r + 2m + 2 = 0$, and further reduces to $r + 2m + 2 = 0$.
| ## Step 4: Expand and simplify the equation from Step 3
Expanding the right-hand side of the equation yields $mr + 2 = mr - 2m + r - 2$, which simplifies to $mr - mr + r + 2 = -2m + r - 2$, or $r - 2m + 2 = -2m + r - 2$.
| ## Step 4: Expand and rearrange the equation to get a quadratic equation in m
Expanding the equation and rearranging the terms, we have $mr + 2 = mr - 2m + r - 2$. Rearranging, we get $2m + 4 = r$. This can be further written as $m + 2 = \frac{r}{2}$ and then $2(m + 2) = r$. Then substitute the new $r$ expression into... | 1 | 0.829727 | 0.743984 | 0.652871 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 5: Express n in terms of m
Substituting the expression $r = 4m + 2$ into the equation $n = mr + 2$, we obtain $n = m(4m + 2) + 2 = 4m^2 + 2m + 2$. | ## Step 5: Substitute the expression for r into the initial equation
Substituting the expression for $r$ into the initial equation, we get $n = m(4m + 2) + 2$.
| ## Step 5: Substitute r in the equation for n
Now we substitute the expression for $r$ from Step 4 into the equation for $n$ from Step 1: $n = (4m + 2 - 2) = 4m$.
| ## Step 5: Substitute the expression for r into the first expression for n and simplify
Substitute the expression for $r$ into the first expression for $n$, giving us $n = 4m + 2) + 2$. Simplifying, we get $n = 4m + 4$.
| 1 | 0.703652 | 0.693242 | 0.631504 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 6: Determine the constraints on m and n
Since there are less than 100 band members, $n < 100$. We need to find the largest integer value of $n$ that satisfies this condition. | ## Step 6: Find the largest number of members the band could have
Since $n < 100$, we can set up an inequality $4m^2 + 2m + 2 < 100$. To find the largest possible value of $m$, we can try values for $m$ such that the expression is closest to 100. It turns out that $m = 6$ results in a large value of $n$ that is less th... | ## Step 6: Find the largest possible n
Since the total number of band members $n$ must be less than 100 and is of the form $4m^2 + 2m + 2$, we need to find the largest value of $m$ for which $n$ is still less than 100. Trying $m = 4$, we find that $n = 4(4)^2 + 2(4) + 2 = 72$. Trying $m = 5$, we find that $n = 4(5)^2 +... | ## Step 6: Find the largest possible value of n with m an integer
To maximize $n$, we need to find the largest value of $m$ such that $n$ is less than 100. Trying out integer values of $m$, we find that $m = 6$ gives $n = 4(6)^2 + 2(6) + 2 = 148$, which is too large. However, $m = 5$ gives $n = 4(5)^2 + 2(5) + 2 = 102$... | 1 | 0.545225 | 0.410039 | 0.349961 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 7: Find the largest possible integer value of m
The largest possible integer value of $m$ is the largest integer such that $4m^2 + 2m + 2 < 100$. | ## Step 7: Test possible values of m to maximize n
Let's start by trying $m = 6$ and see if we get an integer value of $n < 100$. Substituting $m = 6$ into the equation $n = 4m^2 + 2m + 2$, we get $n = 4(36) + 2(6) + 2 = 144 + 12 + 2 = 158$. Since $n = 158 > 100$, we need to try smaller values of $m$. Trying $m = 5$, w... | ## Step 7: Find the maximum value of n
To maximize $n$, we need to find the largest integer value of $m$ such that $4m^2 + 2m + 2 < 100$. This inequality is equivalent to $4m^2 + 2m - 98 < 0$. We can factorize this inequality as $(2m - 7)(2m + 14) < 0$. Since the product of two terms is negative, one term must be posit... | ## Step 7: Find the largest value of n
To maximize $n = 4m^2 + 2m + 2$, we can start with the largest possible value of $m$ that gives $n < 100$. Trying $m = 5$, we obtain $n = 4(5)^2 + 2(5) + 2 = 102$, which exceeds 100. Trying $m = 4$, we obtain $n = 4(4)^2 + 2(4) + 2 = 78$. Since $n$ is decreasing as $m$ decreases, ... | 1 | 0.513359 | 0.303374 | 0.278062 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 8: Solve the inequality
Solving the inequality $4m^2 + 2m + 2 < 100$, we get $4m^2 + 2m - 98 < 0$. This can be written as $2m^2 + m - 49 < 0$. Using the quadratic formula, we can find the roots of the related equation $2m^2 + m - 49 = 0$ to be $m = \frac{-1 \pm \sqrt{1 + 392}}{4}$. The discriminant is greater t... | ## Step 8: Solve the quadratic inequality
$4m^2 + 2m + 2 < 100$ can be rewritten as $4m^2 + 2m - 98 < 0$. To solve this quadratic inequality, we can factor it as $2(2m^2 + m - 49) < 0$, or $(2m - 7)(m + 7) < 0$. The critical points are $m = 7/2$ and $m = -7$, but since $m$ must be an integer, the largest integer $m$ ca... | ## Step 8: Check for m
Solving the inequality $4m^2 + 2m + 2 < 100$ gives $m^2 + \frac{1}{2}m + \frac{1}{2} < 25$, or equivalently $m^2 + m + \frac{1}{2} < 25$, so $m^2 + m - 49.5 < 0$. The largest integer for which the solution holds true is 6.
| ## Step 8: Test values of m to find the largest possible integer value
We test values of $m$ to find the largest possible integer value such that $4m^2 + 2m + 2 < 100$. We find that when $m = 6$, $n = 4m^2 + 2m + 2 = 4 \cdot 36 + 2 \cdot 6 + 2 = 148 > 100$. However, when $m = 5$, $n = 4m^2 + 2m + 2 = 4 \cdot 25 + 2 \cd... | 1 | 0.48375 | 0.426602 | 0.403633 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 9: Find the largest possible integer value of m
Since $m$ must be a positive integer, the largest possible value of $m$ is 7. | ## Step 9: Check which value of m yields the largest value of n
We calculate $n$ for both values of $m$: when $m = -7$, $n = 4(-7)^2 + 2(-7) + 2 = 98$ and when $m = 7$, $n = 4(7)^2 + 2(7) + 2 = 204$. However, 204 exceeds the limit of 100 members. Therefore, the largest value of n the band can have under these condition... | ## Step 9: Test the valid m values
We need to check whether both values of $m$ produce valid integer values of $n$. We first test $m = 7$. When we substitute $m = 7$ into the expression $n = 4m^2 + 2m + 2$, we get $n = 4(7)^2 + 2(7) + 2 = 4(49) + 14 + 2 = 196 + 14 + 2 = 212$. Since $n$ is not less than 100 for $m = 7$,... | ## Step 9: Check which value of m yields the largest value of n that is less than 100
The two possible values of $m$ are $-7$ and $7$. If $m = -7$, we have $n = 4m^2 + 2m + 2 = 4(-7)^2 + 2(-7) + 2 = 4(49) - 14 + 2 = 196 - 14 + 2 = 184$. However, $n = 184$ is greater than 100, which is not allowed. If $m = 7$, we have $... | 1 | 0.416348 | 0.411563 | 0.406035 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 10: Find the corresponding value of n
Substituting $m = 7$ into the expression $n = 4m^2 + 2m + 2$, we obtain $n = 4(7)^2 + 2(7) + 2 = 4(49) + 14 + 2 = 196 + 14 + 2 = 212$. | ## Step 10: Find the corresponding value of n
The corresponding value of $n$ is obtained by substituting $m = 7$ into the expression for $n$: $n = 4(7)^2 + 2(7) + 2 = 196 + 14 + 2 = \boxed{212}$.
| ## Step 10: Find the largest number of members the band could have
Substituting $m = 7$ into the equation $n = 4m^2 + 2m + 2$, we obtain $n = 4 \cdot 49 + 2 \cdot 7 + 2 = 198 + 14 + 2 = 214$.
| ## Step 10: Find the corresponding value of n
Substituting $m = 7$ into the equation $n = 4m^2 + 2m + 2$, we obtain $n = 4 \cdot 7^2 + 2 \cdot 7 + 2 = 196 + 14 + 2 = \boxed{212}$.
| 1 | 0.251382 | 0.231729 | 0.118081 | 1 | 1 | 1 | 1 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | The final answer is: $\boxed{98}$ | Therefore, the final answer is: $\boxed{212}$. | ## Step 11: Find the largest number of members the band could have
Therefore, the largest number of members the band could have is $\boxed{212}$ | Therefore, the final answer is: $\boxed{212}$. | 1 | 0.227686 | 0.219873 | 0.219668 | 1 | 1 | 1 | 1 |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: The given sequence in the column of squares starting from the number 14 forms an arithmetic sequence.
Let's find the common difference between the numbers in this column. Since 18 is the next term in the column after 14, the common difference can be calculated as the difference between the two numbers divide... | ## Step 1: To solve for $N$, we first need to understand that the sequence of integers in each row and column forms an arithmetic sequence.
## Step 2: We notice that the common difference between the elements in the first column is $4$, since $14 - 18 = -4$ and $18 - 21 = -3$. This common difference suggests a common... | ## Step 1: First, let's analyze the arithmetic sequences present in the row of squares and the two columns of squares.
## Step 2: Looking at the row of squares, we can see that it forms an arithmetic sequence with a common difference. We can find the common difference by taking the difference between consecutive term... | 1 | 0.156768 | 0.149326 | 0.099038 | 1 | 1 | 1 | 1 | |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 2: Find the correct difference for the column of squares.
The column of squares forms an arithmetic sequence, and the difference between 18 and -17 is 35. However, looking back, the difference between 14 and 18 is 4, which suggests that there is a common difference in this column of 4, not -9. To check, let's l... | ## Step 2: Re-evaluate the common difference in the column of squares.
However, given two numbers 14 and 18, the difference is 4 which indicates there's an issue with my previous difference. Upon further evaluation we can see that if the two given terms are 21 and 14, the differences are 7. If that is true then we take... | ## Step 2: Find the correct common difference in the column of squares.
Re-evaluating the column on the left, from 21 to 14 is a decrease of 7. Then, from 14 to 18 is an increase of 4, and then from 18 to N is a decrease of 35, so we must find the difference between N and 18 that increases by 9 from the difference betw... | ## Step 2: Recalculate the difference between the top and the third term in the column of squares.
We notice that the difference of $35$ is the difference between $18$ and $-17$. Since $-17$ is the third term, we can get to it by subtracting the common difference from $18$ three times. Thus, the common difference must ... | 1 | 0.242666 | 0.2329 | 0.231719 | 1 | 1 | 1 | 1 |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 3: Find the value of N using the column of squares.
Since the difference between 18 and -17 is 35, the sequence in the right column would be -17, -13, -9, -5, -1, 3, 7, 11, 15, 19, 23, 27, 31, 35. Since the sequence starts at -17 and the difference between terms is 4, we can find the term in the top row of the ... | ## Step 3: Find the value of N in the right-hand column of squares.
The right-hand column of squares has a common difference of 4, based on the terms given in the problem. To find N, we can add 4 to each term from the bottom up until we reach the top. Starting from the bottom, -17, add 4 to get -13, add 4 to get -9, ad... | ## Step 3: Find the value of N.
Now that we know the correct difference for both sequences, we can find N by continuing the sequence in the right column. The sequence in the right column is -17, N - 3. The difference between consecutive terms is 4, so -17 - 4 would give the term before N - 3, which is -21, giving a com... | ## Step 3: Find the value of N using the known differences.
The numbers in the middle row form an arithmetic sequence as well, with a common difference of 4. We can write an expression for the third term using the formula for an arithmetic sequence: $a_n = a_1 + (n - 1)d$. In this case, $a_1 = 21$, $d = 4$, and $n = 3$... | 1 | 0.210752 | 0.164395 | 0.156582 | 1 | 1 | 1 | 1 |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 4: Find the common difference in the row of squares.
However, looking at the row of squares, the difference from 21 to 14 is 7. We can continue this pattern to find the next number in the row. We subtract 7 from 14 to get 7, subtract 7 from 7 to get -0 (or 0), subtract 7 from 0 to get -7, and subtract 7 from -7... | ## Step 4: Find the value of N.
However, looking back at the pattern, we see that 21, 18, and 14 form an arithmetic sequence with a difference of -4. Using this difference, we should be able to go down 4 from 21 to find the next term, which is 17. We can then go down 4 again to get -3, then down 4 again to get -7. Sinc... | ## Step 4: Calculate the final value of N.
Since the left column has a difference of 7 and the sequence starts at 21, we expect terms to be in the sequence of 7's (21, 14, 7, 0, -7, -14, -21, -28, -35, -42, -49, -56, -63, -70). Since -28 is the term in the left column immediately below -17, the term below -17 in the ri... | ## Step 4: Find the value of N by following the sequence of 4's.
However, it is clear that our sequence of terms that we found (25, 29, 33, 37, 41, 45) are incorrect for the given problem. The given terms are 21 and then a decrease of 7, which should lead us to a different sequence for the numbers in the top row. In ad... | 1 | 0.26207 | 0.251406 | 0.177773 | 1 | 1 | 1 | 1 |
An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ | 6 | ## Step 1: Find the common ratio of the first infinite geometric series.
To find the common ratio of the first series, we divide the second term by the first term: $r_1 = \frac{4}{12} = \frac{1}{3}$.
## Step 2: Find the sum of the first infinite geometric series.
Using the formula for the sum of an infinite geometric ... | ## Step 1: Find the common ratio of the first infinite geometric series.
To find the common ratio of the first series, we divide the second term by the first term: $r_1 = \frac{4}{12} = \frac{1}{3}$.
## Step 2: Find the sum of the first infinite geometric series.
Using the formula for the sum of an infinite geometric ... | ## Step 4: Set up an equation using the sum of the second series.
We know that the sum of the second series is four times that of the first series, so $S_2 = 4S_1 = 4(18) = 72$. Using the sum formula, we get $72 = \frac{12}{1-r_2}$. | ## Step 4: Use the given information to relate the sums of the two series.
It is given that the sum of the second series is four times the sum of the first series, so we can write $S_2 = 4S_1$. Plugging in the values we found for $S_1$ and $S_2$, we get $\frac{12}{1 - \frac{4+n}{12}} = 4 \times 18$.
| ## Step 4: Use the given information about the sum of the second infinite geometric series to set up an equation involving $n$.
We are told that the sum of the second series is four times the sum of the first series, so we can set up the equation: $S_2 = 4S_1$. Plugging in the sum of the first series, we have $\frac{12... | ## Step 4: Use the relationship between the sums of the two series to set up an equation.
We are told that the sum of the second series is four times the sum of the first series, so we have $S_2 = 4S_1$. Substituting the expressions for $S_1$ and $S_2$, we get $\frac{12}{1-r_2} = 4\cdot18$.
| 1 | 0.675176 | 0.657969 | 0.606348 | 1 | 1 | 1 | 1 |
An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ | 6 | ## Step 1: Find the common ratio of the first infinite geometric series.
To find the common ratio of the first series, we divide the second term by the first term: $r_1 = \frac{4}{12} = \frac{1}{3}$.
## Step 2: Find the sum of the first infinite geometric series.
Using the formula for the sum of an infinite geometric ... | ## Step 1: Find the common ratio of the first infinite geometric series.
To find the common ratio of the first series, we divide the second term by the first term: $r_1 = \frac{4}{12} = \frac{1}{3}$.
## Step 2: Find the sum of the first infinite geometric series.
Using the formula for the sum of an infinite geometric ... | ## Step 5: Substitute the expression for $r_2$ from Step 3 into the equation from Step 4 and solve for $n$.
Substituting the expression for $r_2$, we get $72 = \frac{12}{1-\frac{4+n}{12}}$. Simplifying this equation will allow us to solve for $n$. | ## Step 5: Solve for $r_2$ from the equation.
We substitute $r_2 = \frac{4+n}{12}$ into the equation $72 = \frac{12}{1-r_2}$. This leads to $72 = \frac{12}{1-\frac{4+n}{12}} = \frac{12}{\frac{12-4-n}{12}} = \frac{144}{8-n}$.
Now solve for $r_2$ to get $\frac{144}{8-n} = 72 \Rightarrow 8-n = 2 \Rightarrow n = -6$.
## St... | ## Step 5: Solve for the common ratio of the second series.
From the previous step, we have $72 = \frac{12}{1-r_2}$. We can solve for $1-r_2$ as follows: $72 = \frac{12}{1-r_2} \Rightarrow 72(1-r_2) = 12 \Rightarrow 60 = 12(1-r_2) \Rightarrow 5 = 1-r_2 \Rightarrow 6 = r_2$.
| ## Step 5: Solve the equation from Step 4 to find the value of $r_2$.
Substituting the expression for $r_2$ from Step 3 into the equation from Step 4 gives $72 = \frac{12}{1-\frac{4+n}{12}}$. Multiplying both sides by the denominator, we get $72\left(1-\frac{4+n}{12}\right) = 12$. Expanding and simplifying the left-han... | 1 | 0.706328 | 0.694004 | 0.581328 | 1 | 1 | 1 | 1 |
Expand the product ${3(x+4)(x+5)}$. | 3x^2 + 27x + 60 | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$.
## Step 2: Apply the distributive property
Apply the distributive property (also known as FOIL) to $(x+4)(... | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$. | ## Step 1: Distribute the 3 across the first term
Distribute 3 to both terms inside the first parenthesis: $3(x+4)(x+5) = 3x(x+5)$.
| ## Step 1: Distribute the factor 3 to the first two binomials
Using the distributive property, we multiply 3 by each term in (x+4) and (x+5):
3(x+4)(x+5) = 3(x+4)(x) + 3(x+4)(5).
| ## Step 1: Multiply the constants and the first binomial
Using the distributive property to expand, we get $3(x+4)(x+5) = 3(x+4)x + 3(x+4)5$.
| 1 | 0.135652 | 0.131726 | 0.125398 | 1 | 1 | 1 | 1 | |
Expand the product ${3(x+4)(x+5)}$. | 3x^2 + 27x + 60 | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$.
## Step 2: Apply the distributive property
Apply the distributive property (also known as FOIL) to $(x+4)(... | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$.
| ## Step 2: Apply the distributive property
Apply the distributive property (also known as FOIL) to $(x+4)(x+5)$:
$(x+4)(x+5) = x(x) + x(5) + 4(x) + 4(5)$
$(x+4)(x+5) = x^2 + 5x + 4x + 20$
$(x+4)(x+5) = x^2 + 9x + 20$ | ## Step 2: Expand $(x+4)(x+5)$ using the FOIL method
Using the FOIL method, we multiply the first terms: $x*x = x^2$.
Then multiply the outer terms: $x*5 = 5x$.
Next, multiply the inner terms: $4*x = 4x$.
Lastly, multiply the last terms: $4*5 = 20$.
| ## Step 2: Multiply the first terms
$(x+4)(x+5) = x \cdot x + x \cdot 5$
| ## Step 2: Apply the distributive property to the first binomial
We'll apply the distributive property to the first binomial, $x + 4$, multiplying each term by $x + 5$.
| 1 | 0.430947 | 0.395352 | 0.348477 | 1 | 1 | 1 | 1 |
Expand the product ${3(x+4)(x+5)}$. | 3x^2 + 27x + 60 | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$.
## Step 2: Apply the distributive property
Apply the distributive property (also known as FOIL) to $(x+4)(... | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$.
## Step 2: Apply the distributive property
Apply the distributive property (also known as FOIL) to $(x+4)(... | ## Step 3: Multiply the result by the third factor
Now multiply the result by the third factor 3:
$3(x^2 + 9x + 20) = 3x^2 + 27x + 60$ | ## Step 3: Multiply the expanded binomial by 3
Now, we'll multiply the expanded binomial by 3:
$3(x+4)(x+5) = 3(x^2 + 9x + 20)$
| ## Step 3: Distribute the last term 3 to the expanded expression of $(x+4)(x+5)$
Next, we'll distribute the factor $3$ to the expanded expression of $(x+4)(x+5)$.
| ## Step 3: Multiply the result by the remaining binomial
Now, we multiply the result from Step 2 by the remaining binomial $(x+5)$ and then multiply the whole result by 3:
$3(x^2 + 9x + 20)$
$3x^2 + 27x + 60$
| 1 | 0.698516 | 0.678047 | 0.425444 | 1 | 1 | 1 | 1 |
Expand the product ${3(x+4)(x+5)}$. | 3x^2 + 27x + 60 | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$.
## Step 2: Apply the distributive property
Apply the distributive property (also known as FOIL) to $(x+4)(... | ## Step 1: Use the distributive property to expand the product of the first two binomials
To expand the product ${3(x+4)(x+5)}$, we'll first use the distributive property on the first two binomials: $(x+4)(x+5)$.
## Step 2: Apply the distributive property
Apply the distributive property (also known as FOIL) to $(x+4)(... | Therefore, the final answer is: $\boxed{3x^2 + 27x + 60}$. | Therefore, the final answer is: $\boxed{3x^2 + 27x + 60}$. | Therefore, the final answer is: $\boxed{3x^2 + 27x + 60}$. I hope it is correct. | Therefore, the final answer is: $\boxed{3x^2 + 27x + 60}$. I hope it is correct. | 1 | 0.784336 | 0.770664 | 0.770664 | 1 | 1 | 1 | 1 |
The center of the circle with equation $x^2+y^2=4x+12y-39$ is the point $(h,k)$. What is $h+k$? | 8 | ## Step 1: Complete the square for the x terms and y terms in the circle equation.
Rearranging and completing the square for the x terms: $x^2-4x$ becomes $(x-2)^2-4$.
Rearranging and completing the square for the y terms: $y^2-12y$ becomes $(y-6)^2-36$.
## Step 2: Rewrite the circle equation using the completed squar... | ## Step 1: Complete the square for the x terms and y terms in the circle equation.
Rearranging and completing the square for the x terms: $x^2-4x$ becomes $(x-2)^2-4$.
Rearranging and completing the square for the y terms: $y^2-12y$ becomes $(y-6)^2-36$.
## Step 2: Rewrite the circle equation using the completed squar... | ## Step 3: Simplify the equation and identify the center of the circle.
Rearrange the equation to group like terms: $(x-2)^2+(y-6)^2=1$.
From the simplified equation, we see the center of the circle is at $(2,6)$. | ## Step 3: Simplify the equation by adding 40 to both sides.
$(x-2)^2+(y-6)^2=40$.
| ## Step 3: Combine like terms and isolate the constant term.
Combine the constant terms on the left side to get: $(x-2)^2+(y-6)^2=4+36+39=79$.
| ## Step 3: Simplify the equation to standard form for a circle.
Combine constants: $(x-2)^2+(y-6)^2-40=-39$.
Add 40 to both sides: $(x-2)^2+(y-6)^2=1$.
Therefore, the final answer is: $\boxed{4}$. | 1 | 0.812969 | 0.811016 | 0.516328 | 1 | 1 | 1 | 1 |
We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$? | 4 | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
## Step 2: Expand and simplify the equation.
Expandin... | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
| ## Step 2: Expand and simplify the equation.
Expanding the product on the left-hand side, we get $2x^2 + 5x - 4x - 10 = 8x - 6$. Combining like terms yields $2x^2 + x - 10 = 8x - 6$. | ## Step 2: Expand the equation and solve for $x$.
Expanding the left side of the equation gives $2x^2 + 5x - 4x - 10 = 8x - 6$. Simplifying, we get $2x^2 + x - 10 = 8x - 6$. Rearranging the terms, we get $2x^2 - 9x + 4 = 0$.
| ## Step 2: Expand the equation and solve for $x$.
Expanding the left-hand side of the equation gives $2x^2 + 5x - 4x - 10 = 8x - 6$. Simplifying further gives $2x^2 + x - 10 = 8x - 6$. Rearranging the terms, we get $2x^2 - 9x + 4 = 0$.
| ## Step 2: Expand and simplify the equation.
Expand the left-hand side of the equation using the distributive property: $2x^2 + 5x - 4x - 10 = 8x - 6$. Combine like terms: $2x^2 + x - 10 = 8x - 6$.
| 1 | 0.763555 | 0.75248 | 0.684668 | 1 | 1 | 1 | 1 |
We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$? | 4 | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
## Step 2: Expand and simplify the equation.
Expandin... | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
## Step 2: Expand and simplify the equation.
Expandin... | ## Step 5: Simplify the expression.
Simplifying the expression inside the square root, we get: $x = \frac{7 \pm \sqrt{81}}{4} = \frac{7 \pm 9}{4}$. | ## Step 5: Simplify and find the solutions for $x$.
This simplifies to $x = \frac{7 \pm \sqrt{81}}{4} = \frac{7 \pm 9}{4}$. This gives us two potential solutions: $x = \frac{7 + 9}{4} = \frac{16}{4} = 4$ and $x = \frac{7 - 9}{4} = -\frac{2}{4} = -\frac{1}{2}$.
| ## Step 5: Simplify the expression.
Simplifying the expression under the square root, we have $x = \frac{7 \pm \sqrt{81}}{4}$. Since $\sqrt{81} = 9$, we have $x = \frac{7 \pm 9}{4}$. Therefore, we have two possible solutions: $x = \frac{7 + 9}{4} = \frac{16}{4} = 4$ or $x = \frac{7 - 9}{4} = \frac{-2}{4} = -0.5$.
| ## Step 5: Simplify the expression and find the solutions.
Simplifying under the square root, we have: $x = \frac{7 \pm \sqrt{81}}{4}$. This further simplifies to $x = \frac{7 \pm 9}{4}$. So, we have two possible solutions for $x$: $x_1 = \frac{7 + 9}{4} = 4$ and $x_2 = \frac{7 - 9}{4} = -\frac{1}{2}$.
| 1 | 0.801855 | 0.800547 | 0.788926 | 1 | 1 | 1 | 1 |
We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$? | 4 | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
## Step 2: Expand and simplify the equation.
Expandin... | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
## Step 2: Expand and simplify the equation.
Expandin... | ## Step 6: Find the possible values of $x$.
We have two possible values for $x$: $x = \frac{7 + 9}{4} = \frac{16}{4} = 4$ and $x = \frac{7 - 9}{4} = \frac{-2}{4} = -\frac{1}{2}$. | ## Step 6: Solve for both possible values of $x$.
Therefore, we have two possible values for $x$: $x_1 = \frac{7 + 9}{4} = \frac{16}{4} = 4$ and $x_2 = \frac{7 - 9}{4} = \frac{-2}{4} = -\frac{1}{2}$.
| ## Step 6: Solve for both possible values of $x$.
So, we have two possible values: $x_1 = \frac{7 + 9}{4}$ and $x_2 = \frac{7 - 9}{4}$. Calculating these values, we get: $x_1 = \frac{16}{4} = 4$ and $x_2 = \frac{-2}{4} = -\frac{1}{2}$.
| ## Step 6: Solve for $x$.
We now have two possible solutions: $x_1 = \frac{7 + 9}{4} = \frac{16}{4} = 4$ and $x_2 = \frac{7 - 9}{4} = -\frac{2}{4} = -\frac{1}{2}$. However, we must check if both solutions satisfy the original conditions of the problem.
| 1 | 0.591895 | 0.591895 | 0.578223 | 1 | 1 | 1 | 1 |
We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$? | 4 | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
## Step 2: Expand and simplify the equation.
Expandin... | ## Step 1: Write the equation for the area of a rectangle.
The area of a rectangle is given by the product of its length and width. In this case, the area is $8x - 6$ and the dimensions are $x - 2$ and $2x + 5$. So, we have the equation: $(x - 2)(2x + 5) = 8x - 6$.
## Step 2: Expand and simplify the equation.
Expandin... | ## Step 7: Check for extraneous solutions.
Given that the dimensions of the rectangle cannot be negative, we must check that $x - 2 > 0$ and $2x + 5 > 0$ for both possible values of $x$. For $x = 4$, we have $4 - 2 = 2 > 0$ and $2(4) + 5 = 13 > 0$. For $x = -\frac{1}{2}$, we have $-\frac{1}{2} - 2 = -\frac{5}{2} < 0$ a... | ## Step 7: Check if the solutions are valid.
We need to check if $x = 4$ and $x = -\frac{1}{2}$ are valid solutions for the given problem. If $x = 4$, then the dimensions are $4 - 2 = 2$ and $2 \cdot 4 + 5 = 13$. The area is $2 \cdot 13 = 26$, not $8 \cdot 4 - 6 = 26$. Therefore, $x = 4$ is not a solution. If $x = -\fr... | ## Step 7: Check if the possible values of $x$ satisfy the conditions of the problem.
For $x = 4$, the dimensions are $4 - 2 = 2$ by $2(4) + 5 = 13$, and the area is $2 \times 13 = 26$. Since $26 \neq 8x - 6 = 8(4) - 6 = 34 - 6 = 28$, $x = 4$ is not a valid solution.
| ## Step 7: Check the possible values of $x$.
We need to check if both possible values of $x$ satisfy the original problem. We'll plug in both values and see if the area of the rectangle is $8x - 6$ for both cases. For $x = 4$, we get an area of $8(4) - 6 = 32 - 6 = 26$, which is not equal to $8x - 6$. For $x = -\frac{1... | 1 | 0.616484 | 0.549199 | 0.489629 | 1 | 1 | 1 | 1 |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | -3 | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex. | ## Step 1: The given graph represents a quadratic function of the form $y=q(x)=ax^2+bx+c$.
## Step 2: We are given a point (7, 4) that lies on the graph of the quadratic function, which means that $q(7) = 4$.
## Step 3: From the graph, we also observe that the axis of symmetry is the vertical line $x=7.5$.
## Step ... | ## Step 1: The graph of a quadratic function is a parabola, and since $q(7)=q(8)$ and $q(6)=q(9)$, it suggests symmetry in the graph about the line $x=7.5$.
## Step 2: Considering the symmetry and the fact that the graph appears to be symmetric about the line $x=7.5$, the value of $q(15)$ is likely to be the same as ... | ## Step 1: The given graph represents a quadratic function in the form of $y=q(x)=ax^2+bx+c$.
## Step 2: We are asked to find the value of $q(15)$, given that it's an integer. Since the graph is symmetric about a vertical line passing through its vertex, we can look for the symmetry of the graph.
## Step 3: Notably,... | 1 | 0.215967 | 0.188477 | 0.007019 | 1 | 1 | 1 | 1 | |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | -3 | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
| ## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$. | ## Step 2: Identify symmetry
The graph shows symmetry with respect to the line $x=7.5$. This means that for every point $(x, y)$ on one side of the line $x=7.5$, there is a corresponding point $(14-x, y)$ on the other side.
| ## Step 2: Identify symmetry
Since the graph exhibits symmetry about the line $x = 7.5$, we can deduce that the axis of symmetry of the parabola is $x = 7.5$. This symmetry property implies that for any point $(a, b)$ on the graph, the point $(14 - a, b)$ is also on the graph.
| ## Step 2: Identify the vertex and its relation to the function
The vertex of the parabola appears to be at $(7.5, 4)$. This is an important point because it tells us that $q(7.5) = 4$.
| 1 | 0.484316 | 0.465488 | 0.439004 | 1 | 1 | 1 | 1 |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | -3 | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
| ## Step 3: Recognize the parabola's symmetry
Since the parabola is symmetrical about the vertical line $x=7.5$, the value of $q(7.5)$ is the same as the value of $q(8)$ and $q(6)$. | ## Step 3: Determine the axis of symmetry
Since the graph is symmetrical about the vertical line $x=7.5$, the axis of symmetry is the vertical line $x=7.5$.
| ## Step 3: Recognize the translation
Since the value of $q(15)$ lies to the right of the vertex, we expect the value to be the same as the value of $q(7)$ but shifted downward. Thus, we expect to get a negative translation from this process.
| ## Step 3: Determine the relation of q(15) to the vertex
Since $q(15)$ is $8$ units to the right of the vertex's $x$-coordinate, its value will be the same as the $y$-coordinate of the vertex minus a multiple of the vertical translation (which is caused by the slope of the parabola). Given the symmetry, q(6) would be 2... | 1 | 0.387363 | 0.37832 | 0.364648 | 1 | 1 | 1 | 1 |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | -3 | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 4: Determine q(7.5)
The value of $q(7.5)$ can be determined by substituting $x=7.5$ into the equation $y=q(x)$. However, we can also determine this by noting that $q(7)$ is $4$ and $q(8)$ is $0$. | ## Step 4: Analyze the relationship between $q(6)$, $q(7.5)$, and $q(15)$
The value of $q(15)$ is the same as the value of $q(7.5)$ and $q(6)$. But since the parabola is symmetrical about the vertical line $x=7.5$, the value of $q(6)$ should be 6 units away from the vertex in the $x$ direction and $q(15)$ should be the... | ## Step 4: Deduce the value of $q(15)$
The graph also shows that $q(0)=-3$ and the vertex is at $(7, 4)$. Hence, $q(15)$ can be evaluated as $q(0)+ 15 \cdot 2 (q(7.5)-q(0))$, where $2 (q(7.5)-q(0)) = q(15) - q(0)$ due to the symmetry.
| ## Step 4: Identify the key feature
The point $(8, 0)$ lies on the graph of $y=q(x)$, and since the graph is symmetrical about $x=7.5$, we also have $q(15)=q(3)=0$.
| 1 | 0.324648 | 0.31623 | 0.216636 | 1 | 1 | 1 | 1 |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | -3 | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 5: Determine q(15)
Since the parabola is symmetrical, $q(15)=q(0)$. | ## Step 5: Find q(15)
To find the value of $q(15)$, note that it is the same distance to the right of the vertex as the vertex is from $x=0$. Since the value of $q(0)$ can be seen to be $q(6)=q(8)$. By adding $8$ to $q(8)$ we get $q(15)$.
| ## Step 5: Calculate q(15)
Since $q(15)$ is at the same height as $q(0)$, we need to find the value of $q(0)$. As $q(7)$ is 4 units above the x-axis and $q(8)$ is 0 units above the x-axis and there is an 8 unit width, then the value of $q(0)$ would be $4-(0-7)/8\cdot4$, which simplifies to $4-(0-7)/8\cdot4=4-(-7/8)\cdo... | ## Step 5: Determine the value of q(15)
Since $q(7)$ and $q(8)$ have integer values and the graph shows symmetry about $x=7.5$, the vertical distance between $q(8)$ and $q(6)$ must be equal to the vertical distance between $q(7)$ and $q(5)$. This indicates that the integer values repeat at an interval of $1$. Therefore... | 1 | 0.216577 | 0.200469 | 0.112944 | 1 | 1 | 1 | 1 |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | -3 | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 1: Analyze the graph
The graph of the quadratic function $y=q(x)$ is a parabola that opens downwards, as indicated by its vertex.
## Step 2: Identify the vertex
The vertex of the parabola appears to be at the point $(7, 4)$, since the graph is symmetrical about the vertical line $x=7.5$.
## Step 3: Recognize ... | ## Step 6: Determine q(0)
The value of $q(0)$ can be determined by observing that the parabola passes through the point $(0,-3)$. | ## Step 6: Find q(0)
Given that $q(7)=4$ and $q(8)=0$, it can be inferred that $q(6)=2$, $q(5)=4$, and $q(4)=6$. Furthermore, the parabola is symmetrical, meaning that the number of units from $q(4)$ to $q(0)$ will be equivalent to the number of units from $q(8)$ to $q(4)$.
## Step 7: Symmetry equality
This, combined w... | ## Step 6: Find q(0)
The value of $q(0)$ can be determined by drawing a horizontal line from the vertex $(7,4)$ down to the $x$-axis at $x=0$. Since the value of $y=q(x)$ is directly proportional to the distance along the $y$-axis, we can calculate $q(0)$ by adding the value of $q(7)$, $q(6)$ and $q(8)$ and dividing by... | ## Step 6: Calculate q(0)
By symmetry, since $q(8)=0$ and the parabola is symmetrical about $x=7.5$, then $q(15) = q(0)$, and $q(0)$ is the reflection of $q(8)$ across the vertical line $x=7.5$. Therefore, $q(15) = -(q(8))=-(0)=\boxed{0}$.
| 1 | 0.356992 | 0.337266 | 0.101839 | 1 | 1 | 1 | 1 |
Evaluate $99\times 99$ in your head. | 9801 | ## Step 1: Break down 99 into easier-to-handle numbers
We can break down 99 as (100 - 1).
## Step 2: Apply the difference of squares formula
The formula for (a - b)^2 is a^2 - 2ab + b^2. In this case, a = 100 and b = 1. Therefore, we can use this formula to expand (100 - 1)^2.
## Step 3: Calculate the expanded form
A... | ## Step 1: Break down 99 into easier-to-handle numbers
We can break down 99 as (100 - 1). | ## Step 1: Calculate the square of 100
$100^2 = 10000$
| ## Step 1: Break down 99 into (100 - 1)
We know 99 is just one less than 100, so we can write $99 = 100 - 1$.
| ## Step 1: Estimate the product by calculating $100\times 100$ and adjusting for the difference.
$100\times 100=10,000$.
| 1 | 0.690801 | 0.681504 | 0.675176 | 1 | 1 | 1 | 1 | |
Evaluate $99\times 99$ in your head. | 9801 | ## Step 1: Break down 99 into easier-to-handle numbers
We can break down 99 as (100 - 1).
## Step 2: Apply the difference of squares formula
The formula for (a - b)^2 is a^2 - 2ab + b^2. In this case, a = 100 and b = 1. Therefore, we can use this formula to expand (100 - 1)^2.
## Step 3: Calculate the expanded form
A... | ## Step 1: Break down 99 into easier-to-handle numbers
We can break down 99 as (100 - 1).
| ## Step 2: Apply the difference of squares formula
The formula for (a - b)^2 is a^2 - 2ab + b^2. In this case, a = 100 and b = 1. Therefore, we can use this formula to expand (100 - 1)^2. | ## Step 2: Use the difference of squares formula
The difference of squares formula is (a - b)(a + b) = a^2 - b^2. Here, we have (100 - 1)(100 + 1) = 100^2 - 1^2.
| ## Step 2: Apply the difference of squares formula
The difference of squares formula is (a - b)(a + b) = a^2 - b^2. Here, a = 100 and b = 1. So, we have (100 - 1)(100 + 1) = 100^2 - 1^2.
| ## Step 2: Use the difference of squares formula to simplify the calculation
The difference of squares formula is (a - b)(a + b) = a^2 - b^2. We can apply this to (100 - 1)(100 + 1) = 100^2 - 1^2 = 9999.
| 1 | 0.679922 | 0.616484 | 0.567373 | 1 | 1 | 1 | 1 |
Given that $x+y = 10$ and $2x+y = 13$, evaluate $x^2-y^2$. | -40 | ## Step 1: Subtract the first equation from the second equation to eliminate y
$(2x+y) - (x+y) = 13 - 10$
This simplifies to $x = 3$.
## Step 2: Substitute the value of x into the first equation to find the value of y
$x + y = 10$
$3 + y = 10$
Subtract 3 from both sides to get $y = 7$.
## Step 3: Evaluate $x^2 - y^2$... | ## Step 1: Subtract the first equation from the second equation to eliminate y
$(2x+y) - (x+y) = 13 - 10$
This simplifies to $x = 3$.
## Step 2: Substitute the value of x into the first equation to find the value of y
$x + y = 10$
$3 + y = 10$
Subtract 3 from both sides to get $y = 7$.
| ## Step 3: Evaluate $x^2 - y^2$
$x^2 - y^2 = (x + y)(x - y)$
We know $x = 3$ and $y = 7$, so $(x + y) = 10$ and $(x - y) = 3 - 7 = -4$.
Therefore, $x^2 - y^2 = (x + y)(x - y) = 10 * -4 = -40$. | ## Step 3: Calculate $x^2-y^2$ using the values of x and y
$x^2 - y^2 = x^2 - (x \times y)$
$x^2 - y^2 = (3)^2 - (3 \times 7)$
$x^2 - y^2 = 9 - 21$
$x^2 - y^2 = -12$
| ## Step 3: Evaluate the expression $x^2 - y^2$
Since $x^2 - y^2 = (x+y)(xy)$, we can substitute the values of $x+y$ and $xy$ into the expression.
$x^2 - y^2 = (x+y)(xy) = (10)(-4)$.
| ## Step 3: Evaluate the expression x^2 - y^2
$x^2 - y^2 = (x+y)(xy)$
Substitute the values of x and y to get $x^2 - y^2 = (10-3)(3-7)$
| 1 | 0.791934 | 0.739824 | 0.558896 | 1 | 1 | 1 | 1 |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? | 8192 | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 1: Identify the sequence pattern
The given sequence starts with 1, and each successive term is the sum of all the previous terms. This sequence is a well-known sequence called the Fibonacci sequence where each term after the first two is the sum of the previous two terms.
| ## Step 1: Understand the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. So, it's a sequence of powers of 2 (2^1 = 1, 2^2 = 2 * 1, 2^3 = 4 * 1 + 2 * 2, ...).
| ## Step 1: Identify the Pattern of the Sequence
The sequence starts with 1, and each successive term is the sum of all the previous terms. This pattern is similar to a well-known sequence: the sequence of powers of 2 (1, 2, 4, 8, ...) and the sequence of Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...) both exhibit simila... | 1 | 0.340176 | 0.319414 | 0.18585 | 1 | 1 | 1 | 1 | |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? | 8192 | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 2: Determine the sequence using powers of 2
Given the sequence description, the nth term can be described as 2^(n-1). Now we need to find the smallest n where 2^(n-1) > 5000. | ## Step 2: Determine which term to check
Since the first term is 1 and we need a term that exceeds 5000, we start with the smallest power of 2 and continue until we find the first term that exceeds 5000.
| ## Step 2: Determine the smallest term in the sequence that exceeds 5000
Since 2^n grows rapidly, we need to find the smallest power of 2 that exceeds 5000. To do this, we take the logarithm of 5000 and look for the smallest integer power of 2 greater than that value.
| ## Step 2: Calculate each term and compare to 5000
We will keep calculating terms of the sequence until we find the first term that exceeds 5000.
| 1 | 0.556016 | 0.542344 | 0.524688 | 1 | 1 | 1 | 1 |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? | 8192 | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 3: Solve for n
We are looking for the smallest n where 2^(n-1) > 5000. To find this, we can test different values of n until we exceed 5000. | ## Step 3: Find n using trial and error on powers of 2
Using powers of 2 and trial and error method:
2^12 = 4096 (less than 5000)
2^13 = 8192 (greater than 5000)
Since 2^(12) is less than 5000 and 2^13 is greater than 5000, the smallest term which exceeds 5000 will be at n = 13.
| ## Step 3: Solve for n
We will find the smallest n where 2^(n-1) is greater than 5000. Starting with n=10, we calculate 2^(10-1) = 2^9 = 512, which is less than 5000. Trying the next higher power of 2, n=11, 2^(11-1) = 2^10 = 1024. However, n=12, 2^(12-1) = 2^11 = 2048 is also less than 5000, and n=13, 2^12 = 4096 is s... | ## Step 3: Find the term that exceeds 5000
Since 2^12 = 4096 and 2^13 = 8192, we can see that 2^12 is less than 5000 but 2^13 is greater than 5000. The first term in the sequence that exceeds 5000 is therefore 2^(13-1) or 2^12, but that is actually the first term that is still below 5000, however 2^13 exceeds 5000. | 1 | 0.64998 | 0.551973 | 0.536348 | 1 | 1 | 1 | 1 |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? | 8192 | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 4: Find the smallest n
Testing values of n:
- 2^(12-1) = 2048, which is less than 5000
- 2^(13-1) = 4096, which is less than 5000
- 2^(14-1) = 8192, which is greater than 5000 | ## Step 4: Calculate and test n values
Testing different values of n, we get:
- 2^(10-1) = 2^9 = 512 (too small)
- 2^(11-1) = 2^10 = 1024 (still too small)
- 2^(12-1) = 2^11 = 2048 (still too small)
- 2^(13-1) = 2^12 = 4096 (still too small)
- 2^(14-1) = 2^13 = 8192 (larger than 5000)
| ## Step 4: Test values of n
Starting from n=10, we find that 2^9 = 512 (which is less than 5000), and 2^10 = 1024 (also less than 5000), and 2^11 = 2048 (still less than 5000), and 2^12 = 4096 (less than 5000), and 2^13 = 8192 (exceeds 5000).
| ## Step 4: Find the specific value of n
Testing different values of n:
- 2^(12-1) = 2048
- 2^(13-1) = 4096
- 2^(14-1) = 8192
8192 is the first value that exceeds 5000. Since this is a sequence of powers of 2, n-1 equals to 14, so n equals to 15.
| 1 | 0.620195 | 0.59627 | 0.551973 | 1 | 1 | 1 | 1 |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? | 8192 | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 5: Determine the term value
Since 2^(14-1) is the first term to exceed 5000, the 14th term of the sequence is the answer. | ## Step 5: Find the nth term
Since 2^(14-1) = 8192 is the first term to exceed 5000, we need to find the 13th and 14th terms of this sequence to confirm the answer, however since 8192 is the first term over 5000, this is our nth term. However we must calculate the value of this term.
| ## Step 5: Identify the term number
Since 2^(14-1) is the first term that exceeds 5000, and it is the 14th term, we look at the 13th term for the answer as the problem asks for the first term to exceed the value of 5000. Therefore, the first term that is greater than 5000 is actually the 14th term but we want the term ... | ## Step 5: Find the corresponding term
We are looking for the 14th term, but we want the term itself (not the term number), so we use n-1 = 13 in our formula. 2^13 equals the 14th term of the sequence minus 1, but to find the 14th term, we calculate 2^13 and add 1.
| 1 | 0.685137 | 0.585371 | 0.229331 | 1 | 1 | 1 | 1 |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? | 8192 | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | ## Step 1: Identify the sequence pattern
The sequence starts with 1, and each successive term is the sum of all the previous terms. This is a well-known sequence known as the "powers of 2 minus 1" or more specifically, the sequence is 1, 1+1 = 2, 1+1+2 = 4, 1+1+2+4 = 8, 1+1+2+4+8 = 16, and so on. However, a more accura... | Therefore, the final answer is: $\boxed{8192}$. | Therefore, the final answer is: $\boxed{8192}$. I hope it is correct. | Therefore, the final answer is: $\boxed{8192}$. I hope it is correct. | Therefore, the final answer is: $\boxed{16384}$. | 1 | 0.809258 | 0.809258 | 0.678398 | 1 | 1 | 1 | 1 |
A car travels the 120 miles from $A$ to $B$ at 60 miles per hour, and then returns to $A$ on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car traveling back from $B$ to $A$? | 36 | ## Step 1: Calculate the time taken to travel from A to B
The time taken is distance divided by speed, so it is 120 miles / 60 mph = 2 hours.
## Step 2: Let's denote the speed of the car traveling back from B to A as x miles per hour
The distance is the same, 120 miles.
## Step 3: Calculate the time taken to travel f... | ## Step 1: Calculate the time taken to travel from A to B
The time taken is distance divided by speed, so it is 120 miles / 60 mph = 2 hours.
| ## Step 2: Let's denote the speed of the car traveling back from B to A as x miles per hour
The distance is the same, 120 miles. | ## Step 2: Calculate the total time taken for the round trip
Let the time taken to travel from B to A be x hours. The average speed is given as 45 mph, so we can use the formula average speed = total distance / total time to solve for total time. The total distance is 240 miles (120 miles each way), so we get: 45 mph =... | ## Step 2: Let's denote the speed from B to A as x miles per hour
We know that the total distance traveled in the round trip is 240 miles. The average speed is the total distance divided by the total time.
| ## Step 2: Calculate the total time taken for the round trip
Let's denote the time taken to travel from B to A as t hours. Since the distance from B to A is the same as from A to B (120 miles), and the average speed is 45 mph, the total time is given by total distance divided by average speed, so it is (120 * 2) / 45 =... | 1 | 0.810039 | 0.780371 | 0.742715 | 1 | 1 | 1 | 1 |
Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel? | 2\sqrt{41} | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$.
#... | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$. | ## Step 1: To find the total distance Brenda needs to travel, we should first calculate the distance from her starting point (-4,5) to the origin (0,0).
The distance formula is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. For this part of the trip, it becomes $\sqrt{(0-(-4))^2 + (0-5)^2}$.
| ## Step 1: First, we need to find the distance from the starting point (-4,5) to the origin (0,0).
To do this, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, which in this case is $d_1 = \sqrt{(0-(-4))^2 + (0-5)^2}$.
| ## Step 1: Find the distance from the starting point (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
For this part of the journey, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$.
Plugging in these values, we get $d = \sqrt{(0-(-4))^2 + (0-5)^2} = ... | 1 | 0.258699 | 0.253159 | 0.240404 | 1 | 1 | 1 | 1 | |
Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel? | 2\sqrt{41} | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$.
#... | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$.
| ## Step 2: Calculate the distance from the origin (0,0) to (5,-4).
For the second part of the trip, $x_1 = 0$, $y_1 = 0$, $x_2 = 5$, and $y_2 = -4$. Using the distance formula again, $d_2 = \sqrt{(5 - 0)^2 + (-4 - 0)^2} = \sqrt{25 + 16} = \sqrt{41}$. | ## Step 2: Calculate the distance from the origin (0,0) to (5,-4).
To find this distance, we again use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the second part of the trip, $x_1 = 0$, $y_1 = 0$, $x_2 = 5$, and $y_2 = -4$. So, $d_2 = \sqrt{(5 - 0)^2 + (-4 - 0)^2} = \sqrt{25 + 16} = \sqrt{41}... | ## Step 2: Calculate the distance from the origin (0,0) to (5,-4).
Using the same distance formula, we have $x_1 = 0$, $y_1 = 0$, $x_2 = 5$, and $y_2 = -4$. Thus, $d_2 = \sqrt{(5 - 0)^2 + (-4 - 0)^2} = \sqrt{25 + 16} = \sqrt{41}$.
| ## Step 2: Calculate the distance from the origin (0,0) to (5,-4).
Using the same distance formula as before, we find the second part of the trip's distance: $x_1 = 0$, $y_1 = 0$, $x_2 = 5$, and $y_2 = -4$. So, $d_2 = \sqrt{(5 - 0)^2 + (-4 - 0)^2} = \sqrt{25 + 16} = \sqrt{41}$.
| 1 | 0.684844 | 0.681953 | 0.678047 | 1 | 1 | 1 | 1 |
Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel? | 2\sqrt{41} | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$.
#... | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$.
#... | ## Step 3: Calculate the total distance traveled by adding the two distances together.
The total distance traveled is the sum of the two individual distances, $d_{total} = d_1 + d_2 = \sqrt{41} + \sqrt{41} = 2\sqrt{41}$. | ## Step 3: Calculate the total distance traveled.
The total distance traveled is the sum of the distances from (-4,5) to the origin and from the origin to (5,-4). Since both parts have the same distance, $d_1 = d_2 = \sqrt{41}$. The total distance is therefore $d_1 + d_2 = \sqrt{41} + \sqrt{41} = 2\sqrt{41}$.
| ## Step 3: Find the total distance traveled.
Since Brenda travels from $(-4,5)$ to the origin and then to $(5,-4)$, we need to find the total distance by adding the two parts of the trip. However, in this case, the distance formula provided equal distances for each segment, so it simplifies to just adding the two dista... | ## Step 3: Calculate the total distance Brenda travels.
Since the distances from (-4,5) to the origin and from the origin to (5,-4) are the same, we can simply add them together or double $d_1$ to get the total distance. Therefore, the total distance $d = 2\sqrt{41}$.
| 1 | 0.791387 | 0.734902 | 0.637109 | 1 | 1 | 1 | 1 |
Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel? | 2\sqrt{41} | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$.
#... | ## Step 1: Calculate the distance from (-4,5) to the origin (0,0).
To find this distance, we use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the first part of the trip, $x_1 = -4$, $y_1 = 5$, $x_2 = 0$, and $y_2 = 0$. So, $d_1 = \sqrt{(0 - (-4))^2 + (0 - 5)^2} = \sqrt{16 + 25} = \sqrt{41}$.
#... | Therefore, the final answer is: $\boxed{2\sqrt{41}}$. | Therefore, the final answer is: $\boxed{2\sqrt{41}}$. I hope it is correct. | Therefore, the final answer is: $\boxed{2\sqrt{41}}$. I hope it is correct. | Therefore, the final answer is: $\boxed{2\sqrt{41}}$. I hope it is correct. | 1 | 0.840898 | 0.840898 | 0.840898 | 1 | 1 | 1 | 1 |
A ball is dropped from 1000 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? | 10 | ## Step 1: Determine the initial height and the pattern of the bounces
The ball is initially dropped from 1000 feet and bounces back up to half the height it just fell.
## Step 2: Calculate the height after each bounce
After the first bounce, the height will be 500 feet, after the second bounce, it will be 250 feet, a... | ## Step 1: Determine the initial height and the pattern of the bounces
The ball is initially dropped from 1000 feet and bounces back up to half the height it just fell.
| ## Step 2: Calculate the height after each bounce
After the first bounce, the height will be 500 feet, after the second bounce, it will be 250 feet, after the third bounce, it will be 125 feet, and so on. We can see that the height is halved after each bounce. | ## Step 2: Calculate the maximum height after each bounce
After the first bounce, the maximum height will be 1000 * 0.5 = 500 feet.
After the second bounce, the maximum height will be 500 * 0.5 = 250 feet.
We continue this pattern until we find the bounce where the maximum height is less than 1 foot.
| ## Step 2: Calculate the height after each bounce until it's less than 1 foot
After the first bounce, the height will be 1000 * 0.5 = 500 feet.
After the second bounce, the height will be 500 * 0.5 = 250 feet.
After the third bounce, the height will be 250 * 0.5 = 125 feet.
After the fourth bounce, the height will be 1... | ## Step 2: Calculate the height after each bounce until it's less than 1 foot
After the first bounce, the height will be 1000 * 0.5 = 500 feet.
After the second bounce, the height will be 500 * 0.5 = 250 feet.
This pattern will continue, with the height being halved after each bounce.
| 1 | 0.60752 | 0.581328 | 0.563574 | 1 | 1 | 1 | 1 |
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