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The sum of the first n terms of the series 1 2 + 3 4 + 7 8 + 15 16 + . is equal to

Options

  1. A2^ n - n -1
  2. B1-2^ -n
  3. C1-2^ -n
  4. D2^ n -1

Correct answer

C. 1-2^ -n

Step-by-step solution

aligned & 1 2 + 3 4 + 7 8 + 15 16 + . . & = (1- 1 2 )+ (1- 1 4 )+ (1- 1 8 )+ (1- 1 16 )+ & =(1+1+1+ . n )- ( 1 2 + 1 4 + 1 8 + 1 16 + . . ) & = n - ( 1 2 + 1 2^2 + 1 2^3 + 1 2^4 + . . ) & = n - 1 2 (1- 1 2^ n ) 1- 1 2 & = n - 1 2 (1- 1 2^ n ) 1 2 & = n -1+2^ - n & =2^ - n + n -1 aligned

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