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  4. The simplified form of in + in +1 + in +2 + in +3 is

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Q. The simplified form of $i^{n} + i^{n +1} + i^{n +2} + i^{n +3}$ is

KCETKCET 2016Complex Numbers and Quadratic Equations

Solution:

We have, $ i^{n} +i^{n+1}+i^{n+2}+i^{n+3} $
$=i^{n}\left(1+i+i^{2}+i^{3}\right) $
$=i^{n}(1+i-1-i)$
$\left[\because i^{2}=-1 \text { and } i^{3}=-i\right] $
$=i^{n} \times 0=0$