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COMEDK2021MathematicsSequences and Series

The sum of the series (1+2)+ (1+2+2² )+ (1+2+2²+2³ )+ upto n terms is

Options

  1. A2^ n+2 -n-4
  2. B2 (2^ n -1 )-n
  3. C2^ n+1 -n
  4. D2^ n+1 -1

Correct answer

A. 2^ n+2 -n-4

Step-by-step solution

The k -th term: T_k = 1 + 2 + 2^2 + + 2^k = 2^ k+1 - 1 S_n = _ k=1 ^ n (2^ k+1 - 1) = _ k=1 ^ n 2^ k+1 - n _ k=1 ^ n 2^ k+1 = 2^2 + 2^3 + + 2^ n+1 = 4 2^n - 1 2-1 = 4(2^n-1) = 2^ n+2 - 4 S_n = 2^ n+2 - 4 - n Correct option: (1)

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