Consider a `G.P.` with first term `(1+x)^(n)`, `|x| lt 1`, common ratio `(1+x)/(2)` and number of terms `(n+1)`. Let `'S'` be sum of all the terms of the `G.P.`, then
The coefficient of `x^(n)` is `'S'` is

Options

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

Answer

A

Explanation

`(a)` `S=((1+x)^(n)[1-((1+x)/(2))^(n+1)])/(1-((1+x)/(2)))`
`=(1)/(2^(n))((1+x)^(n)[2^(n+1)-(1+x)^(n+1)])/(1-x)`
`=(1)/(2^(n))[2^(n+1)(1+x)^(n)-(1+x)^(2n+1)](1-x)^(-1)`
`=(1)/(2^(n))[2^(n+1)(1+x)^(n)-(1+x)^(2n+1)](1+x+x^(2)+...+oo)`
The coefficient of `x^(n)` in `S`
`=(1)/(2^(n))[2^(n+1)sum_(r=0)^(n)'^(n)C_(r )-sum_(r=0)^(n)'^(2n+1)C_(r )]`
`=(1)/(2^(n))[2^(n+1)2^(n)-(1)/(2)2^(2n+1)]`
`=2^(n)`