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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Complex Number

NDA 2021 · Mathematics · Complex Number

What is ∑_ n=1 ^ 8 n+7 i^n equal to, where i= -1 ?

What is ∑n=18n+7in\sum_{n=1}^{8 n+7} i^n equal to, where i=−1i=\sqrt{-1} ?

Options

  1. A-1
  2. B1
  3. Cii
  4. D−i-i

Correct answer

A. -1

Step-by-step solution

i+i2+i3+i4=i−1−i+1=0i+i^2+i^3+i^4=i-1-i+1=0 So value of sum of each concecutive 4 pair is zero. ∴8n+7=8n+4+3=4(2n+1)+3∴∑n=18n+7in=i4(2n+1)+1+i4(2n+1)+2+i4(2n+1)+3=i4(2n+1)[i+i2+i3]=1[i−1−i]=−1\begin{aligned} & \therefore 8 n+7=8 n+4+3=4(2 n+1)+3 \\ & \therefore \sum_{n=1}^{8 n+7} i^n=i^{4(2 n+1)+1}+i^{4(2 n+1)+2}+i^{4(2 n+1)+3} \\ & =i^{4(2 n+1)}\left[i+i^2+i^3\right]=1[i-1-i]=-1 \end{aligned}
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