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[INST] help me answer this question Question: -4 ≤ x ≤ 2, think step by step [/INST] The given inequality is -4 ≤ x ≤ 2. The solutions of this inequality are all real numbers x such that x is greater than or equal to -4 and less than or equal to 2. This can be represented as x ∈ [-4, 2]. , The answer is The solutions of the given equality are all real numbers x greater than or equal to -4 but less than or equal to 2.

[INST] help me answer this question Solve the equation for x: 2x + 3 = 7., think step by step [/INST] We isolate x by subtracting 3 from both sides and then dividing both sides by 2. , The answer is The solution is x = 2.

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question (ii) Subsets of {a, b} are, think step by step [/INST] , The answer is Φ, {a}, {b}, and {a, b}.

[INST] help me answer this question If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer., think step by step [/INST] We have written an as (a – b + b)n and expanded to prove that a – b is a factor of an – bn, whenever n is a positive integer. , The answer is Proved

[INST] help me answer this question The base of an equilateral triangle with side 2a lies along the y-axis, such that the mid-point of the base is at the origin. Find the vertices of the triangle., think step by step [/INST] We can find the coordinates of the vertices by using the properties of equilateral triangles and applying Pythagoras’ theorem. , The answer is The vertices of the triangle are (0, a), (0, -a), (√3a, 0) or (0, a), (0, -a) and (-√3a, 0).

[INST] help me answer this question Find the equation of the line passing through the fixed point (x0, y0), if and only if, the coordinates of (x, y) satisfy the equation y – y0 = m (x – x0)., think step by step [/INST] , The answer is The equation of the line is y – mx = 0.

[INST] help me answer this question A coin is tossed 3 times, and the outcomes are recorded. How many possible outcomes are there?, think step by step [/INST] Since a coin has two outcomes (head or tail) and it is tossed three times, so the total number of possible outcomes is 2*2*2 = 8. , The answer is 8

[INST] help me answer this question Upon comparing, we have (x1 = -2, y1 = 3, z1 = 5); (x2 = 1, y2 = -4, z2 = 6) and (m = 2, n = 3). So, the coordinates of the point which divide the line segment joining the points P (– 2, 3, 5) and Q (1, – 4, 6) in the ratio 2: 3 externally are given by, think step by step [/INST] , The answer is ∴ The coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) are (-8, 17, 3).

[INST] help me answer this question Given, x – y ≤ 2, think step by step [/INST] , The answer is The solution region of the given inequality is above the line x – y = 2. (That is, the origin is included in the region.)

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question Write each of the following statements in the form 'if-then'. (i) You get a job implies that your credentials are good., think step by step [/INST] , The answer is If you get a job, then your credentials are good.

[INST] help me answer this question What is a compound event?, think step by step [/INST] , The answer is An event that has more than one sample point is called a Compound event.

[INST] help me answer this question P (k + 1) is true whenever P (k) is true., think step by step [/INST] Principle of Mathematical Induction , The answer is Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.

[INST] help me answer this question Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = 3 and x = 3 , think step by step [/INST] Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = 3 and x = 3 , The answer is The required points are (0, 3) and (3, 0)

[INST] help me answer this question Find the derivative of the following functions. (vi) 5 sin x – 6 cos x + 7, think step by step [/INST] , The answer is The derivative of 5 sin x – 6 cos x + 7 is not provided.

[INST] help me answer this question Find equation of the line parallel to the line 3x − 4y + 2 = 0 and passing through the point (–2, 3)., think step by step [/INST] The slope of the given line is 3/4. Equation of line having slope m and passing through (x1, y1) is given by y – y1 = m (x – x1). Substituting the given slope and point we get the required equation. , The answer is 3x – 4y = 18

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question The equation of the parabola is of the form y2 = 4ax. The focus is (3, 0), a = 3, think step by step [/INST] , The answer is The equation of the parabola is y2 = 12x

[INST] help me answer this question Which of the following statements are true and which are false? In each case, give a valid reason for saying so.<br>(i) p: Each radius of a circle is a chord of the circle., think step by step [/INST] The given statement is false. Each radius of a circle is not a chord of the circle. A chord of a circle is a line segment whose endpoints lie on the circle, while a radius is a line segment that connects the center of the circle to any point on the circle. , The answer is False.<br>Each radius of a circle is not a chord of the circle.<br>A chord of a circle is a line segment whose endpoints lie on the circle.<br>A radius, on the other hand, is a line segment that connects the center of the circle to any point on the circle.

[INST] help me answer this question Find the indicated term in the sequence whose nth term is an = n2/2n ; a7, think step by step [/INST] On substituting n = 7, we get a7 = 72/27 = 49/128. , The answer is a7 = 49/128

[INST] help me answer this question Women occupy even places, which means they will be sitting in 2nd, 4th, 6th and 8th places, whereas men will be sitting in 1st, 3rd, 5th,7th and 9th places. 4 women can sit in four places and ways they can be seated = 4P4, think step by step [/INST] We have 4 women to be seated in 4 even places. The number of permutations of n objects is nPn = n!. So, the number of ways to seat 4 women in 4 even places is 4P4 = 4! = 4 * 3 * 2 * 1 = 24. , The answer is Ways women can be seated = 24

[INST] help me answer this question The first term of a G.P. is a and the common ratio is r. If the fifth term is 32, and the seventh term is 128, find a and r., think step by step [/INST] , The answer is We have a = a, r = ar^(n–1). Then, a4 = ar^3 = (–3) r^3, a2 = a r^1 = (–3) r. We have (–3) r^3 = [(–3) r]^2 and r = –3, so a7 = a r^7–1 = a r^6 = (–3) (–3)^6 = – (3)^7 = –2187.

[INST] help me answer this question The elements are the letters of the word MATHEMATICS. Hence, (iii) matches with (d)., think step by step [/INST] , The answer is The elements of set (iii) are the letters of the word MATHEMATICS, which makes it match with set (d).

[INST] help me answer this question Convert the angle measure 120 degrees to radians., think step by step [/INST] To convert an angle measure from degrees to radians, we can use the formula: radians = (degrees * π) / 180. Therefore, 120 degrees is equivalent to (120 * π) / 180 = 2.0943951023932 radians. , The answer is The equivalent angle measure in radians is 2.0943951023932 radians.

[INST] help me answer this question What is C ∩ D?, think step by step [/INST] C ∩ D represents the set of elements that are present in both C and D. Here, C is the set of odd natural numbers and D is the set of prime numbers. Therefore, C ∩ D is the set of odd prime numbers. , The answer is {x: x is an odd prime number}

[INST] help me answer this question Expand (5x - 3y + 2)², think step by step [/INST] , The answer is (5x - 3y + 2)² = 25x² - 30xy + 9y² + 20x - 12y + 4

[INST] help me answer this question What is the probability of getting at least one tail when a coin is tossed twice?, think step by step [/INST] , The answer is 3/4

[INST] help me answer this question 1/x > 1/(x + 1), think step by step [/INST] , The answer is x ∈ (-1, 0) ∪ (0, ∞)

[INST] help me answer this question Find the equation of the parabola with equation y^2 = 10x., think step by step [/INST] On comparing the given equation with y^2 = 4ax, we get a = 5/2. The coordinates of the focus are (a,0) = (5/2, 0). The given equation involves y^2, so the axis of the parabola is the x-axis. Therefore, the equation of directrix is x = -a, which is x = -5/2. The length of latus rectum is 4a = 4(5/2) = 10. , The answer is y^2 = 10x.

[INST] help me answer this question Checking for the origin (0, 0) 0 ≤ 30, this is true. Hence, the origin lies in the solution area, which is given by the left side of the line graph. , think step by step [/INST] 0 ≤ 30, this is true. , The answer is The origin lies in the solution area, which is given by the left side of the line graph.

[INST] help me answer this question In the expansion of (1 + a)m+n, prove that coefficients of am and an are equal., think step by step [/INST] , The answer is By substituting the values in the general form, Tr+1 = m+n Cr ar, we can prove that the coefficients of am and an are equal.

[INST] help me answer this question Are (0, 7, 10), (–1, 6, 6), and (–4, 9, 6) vertices of a right-angled triangle?, think step by step [/INST] PQ2 + QR2 = PR2, which shows that PQR is a right-angled triangle at point Q. , The answer is Yes, at point Q

[INST] help me answer this question Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8)., think step by step [/INST] , The answer is Hence, the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8) is 2:3.

[INST] help me answer this question Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9., think step by step [/INST] , The answer is The equation of the line is 2x + y – 6 = 0 or x + 2y – 6 = 0.

[INST] help me answer this question How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9, which are divisible by 10 and no digit is repeated?, think step by step [/INST] , The answer is The number of 6-digit numbers that can be formed is 120.

[INST] help me answer this question If the sum of n terms of an A.P. is 3n^2 + 5n and its mth term is 164, find the value of m., think step by step [/INST] Explanation provided in the markdown text outlines the steps to derive the answer. It includes equating the sum of n terms from both the equations, simplifying the equation, and then solving for m. , The answer is The value of m is obtained by solving an equation in terms of m.

[INST] help me answer this question F ∩ I, think step by step [/INST] I is not defined in the question. , The answer is F ∩ I = (3, 4, 5, 6) ∩ (I) = Not provided

[INST] help me answer this question There are two ways in which vowels and consonants can appear together: (AEIOU) (QTN) or (QTN) (AEIOU). How many total number of ways can vowel and consonant appear together?, think step by step [/INST] , The answer is The total number of ways in which vowel and consonant can appear together are 2 × 5C5 × 3C3 = 1440.

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question Expand (1 + 3x + 2x2)3, think step by step [/INST] , The answer is From the binomial theorem, the given equation can be expanded as Solution: Co (1)3 + 3C1 (1)2 (3x) + 3C1 (1) (3x)2 + 3C1 (1)2 (2x2) + 3C2 (1) (3x) (2x2) + (2x2)3

[INST] help me answer this question Find the sum to n terms of the series 3 × 12 + 5 × 22 + 7 × 32 + …, think step by step [/INST] , The answer is The sum of n terms of the series can be expressed as the sum of the first n terms of the sequence an, where an = 2n3 + n2.

[INST] help me answer this question What is the length of the shortest board?, think step by step [/INST] , The answer is The length of the shortest board is greater than or equal to 8 cm and less than or equal to 22 cm.

[INST] help me answer this question For x + y ≤ 6, x + y ≥ 4, find the region of the solution., think step by step [/INST] , The answer is The region of the solution for the given inequalities is the area on the left side of the line x + y = 6, including the origin, and the area above the line x + y = 4 or the area on the right of the line.

[INST] help me answer this question Find the coordinates of the foci, vertices, length of major axis, length of minor axis, eccentricity, and length of latus rectum for the equation 36x^2 + 4y^2 = 144., think step by step [/INST] , The answer is {'foci': ['(0, 4√2)', '(0, -4√2)'], 'vertices': ['(0, 6)', '(0, -6)'], 'major axis length': 12, 'minor axis length': 4, 'eccentricity': '2√2/3', 'latus rectum length': '4/3'}

[INST] help me answer this question Prove that sin (x - y) = sin x cos y - cos x sin y, think step by step [/INST] N/A , The answer is sin (x - y) = sin x cos y - cos x sin y

[INST] help me answer this question sin 2x + cos x = 0, think step by step [/INST] , The answer is The general solution of this equation is given by cos x = 0 or 2 sin x + 1 = 0, which implies x = nπ/2 or 2x = nπ + 3π/8 where n ∈ Z.

[INST] help me answer this question - 3 ≤ 4 – 7x/2 ≤ 18, think step by step [/INST] , The answer is x ∈ [-4, 2]

[INST] help me answer this question Is it a statement?, think step by step [/INST] All real numbers can be written as a × 1 + 0 × i. , The answer is Yes, it is a statement.

[INST] help me answer this question Show that the given points are the vertices of a parallelogram., think step by step [/INST] , The answer is ABCD is a parallelogram.

[INST] help me answer this question What is the probability of getting exactly two tails?, think step by step [/INST] , The answer is The probability of getting exactly two tails is 3/8.

[INST] help me answer this question Write the first five terms of the sequence whose nth term is an = (2n – 3)/6, think step by step [/INST] , The answer is -1/6, 1/6, 1/2, 5/6, 7/6

[INST] help me answer this question Number of vowels in PERMUTATIONS = 5 (E, U, A, I, O), Now, we consider all the vowels together as one. Number of permutations of vowels = 120, Now total number of letters = 12 – 5 + 1= 8. Find the number of permutations., think step by step [/INST] , The answer is Number of permutations = 8!/4! = 1680

[INST] help me answer this question Evaluate: , think step by step [/INST] , The answer is The value of the expression.

[INST] help me answer this question What makes a collection a set?, think step by step [/INST] A set is a well-defined collection of objects, often referred to as elements or members. A collection is well-defined if the criteria for membership are clear. , The answer is A collection is a set if it is well-defined, and one can find a number which belongs to this collection. A collection is a set if it is well-defined, and one can find a book which belongs to this collection. A collection is a set if it is well-defined, and one can find an integer which belongs to this collection. A collection is a set if it is well-defined, and one can find a question which belongs to this chapter.

[INST] help me answer this question 2(x + 1) + 3x ≤ 8x + 1, think step by step [/INST] , The answer is x ∈ (∞, -3.5)

[INST] help me answer this question 37 – (3x + 5) ≥ 9x – 8 (x – 3), think step by step [/INST] The solutions of the given inequality are defined by all the real numbers less than or equal to 3. Hence, the required solution set is (-∞, 3]. , The answer is x ≤ 3

[INST] help me answer this question Find the value of x for which the points (x, – 1), (2, 1) and (4, 5) are collinear., think step by step [/INST] , The answer is The required value of x is 1.

[INST] help me answer this question What are the required points on the graph for the given inequalities of 5x + 4y ≤ 20?, think step by step [/INST] These points satisfy the given inequalities. , The answer is The required points would be (1, 0), (1, 2) and so on.

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

[INST] help me answer this question Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non-defective (N). Write the sample space of this experiment., think step by step [/INST] N/A , The answer is S = {DDD, DDN, DND, NDD, DNN, NDN, NND, NNN}

[INST] help me answer this question Calculate the standard deviation for the group B data., think step by step [/INST] , The answer is The standard deviation for group B is 15.61.

[INST] help me answer this question What is the probability of getting at most two tails?, think step by step [/INST] , The answer is The probability of getting at most two tails is 7/8.

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

[INST] help me answer this question Find the sum to n terms of the series whose nth term is given by n2 + 2n., think step by step [/INST] , The answer is The sum to n terms of the series can be expressed as (1/3)n(n+1)(n+2).

[INST] help me answer this question Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B., think step by step [/INST] For question 2, the number of subsets of A × B is 2^6 = 64. Therefore, the number of relations from A to B is 64. , The answer is The number of relations from A to B is 2^6 = 64.

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question Reduce the following equations into intercept form and find their intercepts on the axes. (i) 3x + 2y – 12 = 0 (ii) 4x – 3y = 6 (iii) 3y + 2 = 0, think step by step [/INST] The equation of a line in intercept form is x/a + y/b = 1, where a and b are the intercepts on the x and y axes, respectively. To reduce the given equations into intercept form, we need to rearrange them so that x and y are on the left-hand side and the constants are on the right-hand side. We then divide both sides by the constant term to get the intercepts. , The answer is Intercept on x-axis = 4, Intercept on y-axis = 6 Intercept on x-axis = 3/2, Intercept on y-axis = -2 Intercept on x-axis = 0, Intercept on y-axis = -2/3

[INST] help me answer this question It is seen that the same first element, i.e., 1, corresponds to two different images, i.e., 3 and 5; this relation cannot be called a function., think step by step [/INST] , The answer is This relation cannot be called a function because the same first element corresponds to two different images.

[INST] help me answer this question If A (4, 0, 0) and B (–4, 0, 0), find the equation of the set of points P (x, y, z) such that the sum of the distances from A and B is equal to 10., think step by step [/INST] By equating the sum of distances PA and PB to 10 and simplifying, we get the equation x – 2z = 0. , The answer is x – 2z = 0

[INST] help me answer this question What is E ∩ F?, think step by step [/INST] , The answer is The value of E ∩ F is 6.

[INST] help me answer this question How many points are there in the sample space of selecting a card from a pack of 52 cards?, think step by step [/INST] , The answer is 52

[INST] help me answer this question Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them., think step by step [/INST] , The answer is A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} The set A × B has 2^4 = 16 subsets. These are: {Φ}, {(1, 3)}, {(1, 4)}, {(2, 3)}, {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3),(2,3)},{(1,3), (2,4)},{(1,4), (2,3)},{(1,4), (2,4)},{(2,3), (1,4)},{(2,3), (2,4)},{(1,3), (1,4), (2,3)},{(1,3), (1,4), (2,4)},{(1,4), (2,3), (2,4)},{(1,3), (2,3), (1,4), (2,4)}

[INST] help me answer this question Which of the following pairs of sets are disjoint?, think step by step [/INST] See subparts , The answer is Not disjoint

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question Prove that cos (x - y) = cos x cos y + sin x sin y, think step by step [/INST] N/A , The answer is cos (x - y) = cos x cos y + sin x sin y

[INST] help me answer this question Which of the following may be considered as the universal set for all the three sets A, B and C?, think step by step [/INST] , The answer is The set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

[INST] help me answer this question Find the value of x in the given equation: 2x/3 - 1 = 0, think step by step [/INST] , The answer is The value of x is 3/2.

[INST] help me answer this question The Cartesian product A × A has 9 elements, among which are found (–1, 0) and (0, 1). Find the set A and the remaining elements of A × A., think step by step [/INST] , The answer is A = {–1, 0, 1}

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

[INST] help me answer this question The student can choose 3 questions from part I and 5 from part II, Or 4 questions from part I and 4 from part II, Or 5 questions from parts 1 and 3 from part II., think step by step [/INST] This is a combination problem. For each option given, we need to calculate the number of ways to choose the questions from part I and part II. For example, for the first option, there are 5C3 ways to choose 3 questions from 5 in part I and 7C5 ways to choose 5 questions from 7 in part II. To find these combinations, we need to use the formula mentioned in the previous question. , The answer is The student can choose the questions in 56, 175, or 273 ways, respectively

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question What is the value of x in the following equation: 3x - 7 = 20?, think step by step [/INST] To find the value of x, we need to isolate it on one side of the equation. We can do this by adding 7 to both sides of the equation to get 3x = 27, and then dividing both sides by 3 to get x = 9. , The answer is The value of x is 9.

[INST] help me answer this question Question: The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour, and nth hour?, think step by step [/INST] Explanation: Given the situation, the sequence forms a G.P. To solve for the number of bacteria in each given case, we can use the properties of a G.P. , The answer is [{'part': 'At the end of the 2nd hour: 120 bacteria'}, {'part': 'At the end of the 4th hour: 480 bacteria'}, {'part': 'At the end of the nth hour: 30 * 2^n bacteria'}]

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question Given 2x – y >1, think step by step [/INST] Checking for the origin, putting (0, 0) 0 >1, which is false , The answer is The points are (0,-1) and (0.5, 0)

[INST] help me answer this question If x1 = 4, y1 = – 7, z1 = 8, x2 = 2, y2 = – 3, z2 = 4, find the distance CD., think step by step [/INST] The distance CD is calculated using the formula: √[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]. Substituting the given values, we get: √[4 + 16 + 16] = √36 = 6. , The answer is 6

[INST] help me answer this question Evaluate the given limit:, think step by step [/INST] Given limit, substitute x = 1, we get (a*1^2 + b*1 + c) / (c*1^2 + b*1 + a) = 1, given a + b + c = 1 , The answer is 1

[INST] help me answer this question What is the formula used to calculate the standard deviation?, think step by step [/INST] , The answer is σ = √Variance

[INST] help me answer this question cos 3x + cos x – cos 2x = 0, think step by step [/INST] , The answer is The general solution of this equation is given by cos x (2 cos x - 1) = 0, which implies cos x = 0 or 2 cos x - 1 = 0.

[INST] help me answer this question What is the relationship between set B and set D?, think step by step [/INST] , The answer is Set B is equal to set D.

[INST] help me answer this question Find the domain and the range of the real function f defined by f (x) = |x – 1|., think step by step [/INST] , The answer is The domain of the function f(x) = |x – 1| is R (set of all real numbers) and its range is the set of all non-negative real numbers.

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

[INST] help me answer this question What is the sample space for the experiment where a die is thrown repeatedly until a six comes up?, think step by step [/INST] , The answer is The sample space, denoted as S, is infinitely defined and includes all the possible combinations of throws that result in a six, such as {(6), (1,6), (2,6), (3,6), (4,6), (5,6), (1,1,6), (1,2,6)...}.

[INST] help me answer this question If you have winter clothes, then you live in Delhi., think step by step [/INST] This is the converse of the statement in (a). , The answer is

[INST] help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step [/INST] Explaination: N/A , The answer is Answer: Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.