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KVPY2011MathematicsSequences and Series

Suppose n is a natural number such that |i+2 i²+3 i³+ +n i^ n |=18 2 , where i is the square root of -1 . Then n is -

Options

  1. A9
  2. B18
  3. C36
  4. D72

Correct answer

C. 36

Step-by-step solution

S = i +2 i ²+3 i ³+ . .+ ni ^ n i S=i²+2 i³+ . .+(n-1) i^ n +n i^ n+1 S(1-i)=i+i²+i³+ .++i^ n -n i^ n+1 S (1- i )= i (1- i ^ n ) 1- i - ni ^ n +1 S = 1- i ^ n -2 i - ni ^ n + i 1- i z ₁ (say) z ₂ (say) |z₁ |= 1 2 or 0 |z₂ |= n 2 = n 2 2 n 2 =18 n =36

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