1.

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

Answer»

`(3//4)^()`
`1`
`2^(n)`
`3^(n)`

SOLUTION :`(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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