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NDA 2 2024 Maths, question 10

Topic: Algebra

Q10

What is the value of the sum

∑n=120(in−1+in+in+1)\sum _{n=1}^{20}(i^{n-1}+i^{n}+i^{n+1})

where i=−1i=\sqrt{-1}?

  1. (a) −2i-2i
  2. (b) 00Correct answer
  3. (c) 11
  4. (d) 2i2i

Solution.

Answer: (b) 00

  1. Factor each summand as iⁿ⁻¹(1+i+i²)=iⁿ.
  2. The powers i, −1, −i, 1 repeat in groups of four with sum zero.
  3. Twenty terms contain five complete groups, so the total is zero.