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
`sum_(r=0)^(n)"^(n+r)C_(r )((1)/(2))^(r )` equals

Options

A. `(3//4)^()`
B. `1`
C. `2^(n)`
D. `3^(n)`

Answer

C

Explanation

`(c )` `sum_(r=0)^(n)'^(n+r)C_(r )((1)/(2))^(r )`
`=^(n)C_(n)((1)/(2))^(0)+^(n+1)C_(n)((1)/(2))^(1)+^(n+2)C_(n)((1)/(2))^(2)+......+^(2n)C_(n)((1)/(2))^(n)`
`="coefficient of"x^(n)"in" (1+x)^(n)((1)/(2))^(0)+(1+x)^(n+1)((1)/(2))^(1)+(1+x)^(n+2)((1)/(2))^(2)+....+(1+x)^(2n)((1)/(2))^(n)`
`="coefficient of "x^(n)"in"S`
`=2^(n)`