Question

The number of elements in the set $S=\{i^n+i^{-n}$ : $n$ is an integer $\}$ where $i=\sqrt{-1},$ is

• A. $3$
• B. $1$
• C. greater than $4$ but finite
• D. infinite

Answer 1

Answer: (A)

$3$

$i^n+i^{-n}$
$=\frac{i^{2n}+1}{i^n}$
$f\left(n\right)=\frac{(-1{)}^n+1}{i^n}$
Now if, $n$ is multiple of $4$, we get $f\left(n\right)=1$
If $n$ is a multiple of $2$, and not a multiple of, $4$, we get $f\left(n\right)=-1$
And if $n$ is odd, we get $f\left(n\right)=0$
Hence there will be in total $3$, elements, in S.