The sum to infinity of the series
`1+2(1-(1)/(n))+3(1-(1)/(n))^(2)+ . . .. . . .`, is (A)`n^(2)` (B)`n(n+1)` (C)`n(1+(1)/(n))^(2)` (D)None of these

Explanation

(ii) ` therefore S_(infty)=1+4((1)/(5))+7((1)/(5))^(2)+10((1)/(5))^(3)+"...." "upto "infty ".....(iii)"`
Multiplying both sides of Eq. (i) by `(1)/(5)`, we get
`(1)/(5) S_(infty)=((1)/(5))+4((1)/(5))^(2)+7((1)/(5))^(3)+"...." "upto "infty ".....(iv)"`
Subtracting Eq. (iv) from Eq. (iii), we get
`(1-(1)/(5)) S_(infty)=1+3[((1)/(5))+((1)/(5))^(2)+((1)/(5))^(3)+"...." "upto "infty "]`
`=1+3(((1)/(5))/(1-(1)/(5)))=1+(3)/(4)`
`=(4)/(5)S_(infty)=(7)/(4)=`
`therefore S_(infty)=(35)/(16)`