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

Options

A. `2^(n) - n - 1`
B. `1 -2^(-n)`
C. `n -1- 2^(-n) `
D. `2^(n) + 1`

Answer

C

Explanation

Sum of the n terms of the series `(1)/(2) + (3)/(4) + (7)/(8) + (15)/(16) +`.... Upto n terms can be written as
`(1 - (1)/(2)) + (1 - (1)/(4)) + (1 - (1)/(8)) + (1 - (1)/(16)) +`... upto n terms
`= n - ((1)/(2) + (1)/(4) + (1)/(8) + ...+ n" terms ")`
`= n - ((1)/(2) (1 - (1)/(2^(n))))/(1 - (1)/(2)) = n + 2^(-n) -1`