standard_description: Order three objects by length; compare the lengths of two objects indirectly by using a third object.
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
$$
i^{195}+i^{194}+\cdots+i^2+i+1.
$$


aligns: no