1957 AMC 12 Problem 42

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

Powers of

i

repeat with period four

Check one representative of each residue class of

n

modulo

4

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.

Problem 42 in other years