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NDA2024MathematicsComplex NumberActual

What is the value of the sum _ n=1 ²⁰ (i^ n-1 +i^n+i^ n+1 ) where i= -1 ?

Options

  1. A-2 i
  2. B0
  3. C1
  4. D2 i

Correct answer

B. 0

Step-by-step solution

aligned & Suppose A= _ n=1 ²⁰ (i^ n-1 +i^n+i^ n+1 ) & A= _ n=1 ²⁰ i^n ( 1 i +1+i )= _ n=1 ²⁰ i^n(-i+1+i) & = _ n=1 ²⁰ i^n=i^1+i^2+i^3+i^4 . .+i²⁰ & =5 (i+i^2+i^3+i^4 ) & =5 (0)=0 i+i^2+i^3+i^4=0 aligned

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