Question

Sum of the first n terms of the series `1/2 + 3/4 + 7/8 + 15/16 +`......... is equal to:

Options

• `2^n - n - 1`

• `1 - 2^(-n)`

• `n + 2^(-n) - 1`

• `2^n + 1`

Answer 1

`n + 2^(-n) - 1`

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`