जहाँ i = √-1 दिया गया है, तो योग  \(\rm \Sigma_{n=1}^{20}(i^{n-1}+i^n+i^{n+1})\) का मान क्या है?

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NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
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  1. -2i
  2. 0
  3. 1
  4. 2i

Answer (Detailed Solution Below)

Option 2 : 0
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व्याख्या:

माना कि I = \(\rm \Sigma_{n=1}^{20}(i^{n-1}+i^n+i^{n+1})\)

⇒ I = \(\rm \Sigma_{n=1}^{20}(i^n (\frac{1}{i} +1+ i)\)

⇒ I = \(\rm \Sigma_{n=1}^{20}(i^n (1)\)

⇒ I = \(i^1+ i^2+i^3+i^4+i^5+........+ i^{20}\)

⇒ I = \(5.(i^1+ i^2+i^3+i^4) = 0\) ( {∵ i + i2 + i3 + i4 = 0} )

∴ विकल्प (b) सही है।

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