Question

What is the value of i^20+1?

Answer 1

You know

$i^2=-1$

Squaring both sides gives

$(i^2)^2=(-1)^2\implies i^4=1$

Then raising both sides to any integer power $n$ gives

$(i^4)^n=1^n\implies i^{4n}=1$

That is, any power of $i$ where the power is a multiple of 4 will always be equal to 1.

So

$i^{20}+1=1+1=2$

But perhaps you mean to write 20+1 as the exponent, in which case

$i^{20+1}=i^{20}i=i$