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MHT CET202619 April 2026Evening ShiftMathematicsComplex NumberActual

If z = _ n=0 ²⁰²⁶ i^n , where i = -1 , then one of the values of z is...

Options

  1. Ae^ i 4
  2. B1 2 e^ i 4
  3. Ce^ i 2
  4. D1 2 e^ i 2

Correct answer

A. e^ i 4

Step-by-step solution

The given series is a geometric progression with 2027 terms, first term 1 , and common ratio i . Using the sum formula for a geometric progression: z = 1(1 - i²⁰²⁷) 1 - i Since 2027 = 4 506 + 3 , we have i²⁰²⁷ = (i^4)⁵⁰⁶ i^3 = 1 (-i) = -i . Substituting this into the expression for z : z = 1 - (-i) 1 - i = 1 + i 1 - i Multiplying the numerator and denominator by the conjugate of the denominator, 1 + i : z = (1 + i)^2 (1 - i)(1 + i) = 1 + i^2 + 2i 1^2 - i^2 Since i^2 = -1 : z = 1 - 1 + 2i 1 - (-1) = 2i 2 = i In expo

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