Question

If $S=i^n+i^{-n},$ where $i=\sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is:

$1$

$2$

$3$

$4$

more than $4$

Answer: C

Concepts:complex number roots of unity modular arithmetic

Difficulty rating: 1530

Small Hint:

Powers of $i$ repeat with period four

Big Hint:

Check one representative of each residue class of $n$ modulo $4$

Solution:

If $n$ is even, $i^n$ is $1$ or $-1$ and equals its reciprocal, so $S=2$ or $-2.$ If $n$ is odd, the two terms are $i$ and $-i,$ in some order, so $S=0.$ Thus the distinct values are $-2,\ 0,\ 2,$ three values.

Therefore, the correct answer is C.

1957 AMC 12 Problem 42

Answer: C

Difficulty rating: 1530

Small Hint:

Big Hint:

Solution:

Problem 42 in Other Years