COMEDK · Maths · 3. Complex Number
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 \(i^n\)
- D 0
Answer & Solution
Correct Answer
(D) 0
Step-by-step Solution
Detailed explanation
\(\begin{aligned}\text{Given,} i^n+ & i^{n+1}+i^{n+2}+i^{n+3}=i^n\left(1+i+i^2+i^3\right) \\ & =i^n\left(1+i+(-1)+\left(i^2\right) i\right) \\ & =i^n(1+i+(-1)+(-1) i) \\ & =i^n[(1+i)-(1+i)]=i^n(0)=0\end{aligned}\)
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