Q109 / 180 · COMEDK 2023 (Memory Based) · Mathematics

If $i=\sqrt{-1}$ and $n$ is a positive integer, then $i^n+i^{n+1}+i^{n+2}+i^{n+3}$ is equal to

  • A
    1
  • B
    $i$
  • C
    $i^n$
  • Correct answer
    D
    0

Solution

$\begin{aligned}\text{Given,} i^n+ & i^{n+1}+i^{n+2}+i^{n+3}=i^n\left(1+i+i^2+i^3\right) \\ & =i^n\left(1+i+(-1)+\left(i^2\right) i\right) \\ & =i^n(1+i+(-1)+(-1) i) \\ & =i^n[(1+i)-(1+i)]=i^n(0)=0\end{aligned}$

Hence, the correct answer is option: 'D'.

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