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Consider a G.P. with first term (1+x)^(n...

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 (a) `3/4` (b) `1` (c)`2^n` (d) `3^n`

A

`(3//4)^()`

B

`1`

C

`2^(n)`

D

`3^(n)`

Text Solution

Verified by Experts

The correct Answer is:
C
`(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)`
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