options: Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of numbers to the system of rational numbers.

Reason quantitatively and use units to solve problems.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform operations on matrices and use matrices in applications.

Compute fluently with multi-digit numbers and find common factors and multiples.

Perform arithmetic operations with complex numbers.

Perform operations on vectors.

Extend the properties of exponents to rational exponents.

Use properties of rational and irrational numbers.

Represent complex numbers and their operations on the complex plane.

Represent and model with vector quantities.
problem_activity: Task
For this task, the letter $i$ denotes the imaginary unit, that is, $i=\sqrt{-1}$.
For each integer $k$ from 0 to 8, write $i^k$ in the form $a+bi$.
Describe the pattern you observe, and algebraically prove your observation.  In particular, simplify $i^{195}$.
Write each of the following expression in the form $a+bi$:
$i^2 + i + 1$
$i^3 + i^2 + i +1$
$i^4 + i^3 + i^2 + i +1$
$i^5 + i^4 + i^3 + i^2 + i +1$
$i^6 + i^5 + i^4 + i^3 + i^2 + i +1$
$i^7 + i^6 + i^5 + i^4 + i^3 + i^2 + i +1$
$i^8 + i^7 + i^6 + i^5 + i^4 + i^3 + i^2 + i +1$
Describe the pattern you observe, and algebraically prove your observation.  In particular, compute
$$
i^{195}+i^{194}+\cdots+i^2+i+1.
$$


correct_option_index: 8