Question

Evaluate;
1+i^2+i^4+i^6+.....................+i^20

Answer 1

as we know i=-1
i^2=-1
i^4=1
1-1+1-1+1-1+1-1+1-1+1=1

Answer 2

$i = \sqrt{-1} \\ i^2 = -1 \\ i^4 = 1 \\ Odd\ powers\ of\ i^2\ is\ -1 \\ Even\ powers\ of\ i^2\ is\ 1 \\ \\ Answer\ = 1 -1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 -1 + 1 = 1 \\ There\ are\ 5\ odd\ powers\ and \ 5\ even\ powers\ of\ -1.$

answer = 1