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

Options

A. `n^(2)`
B. `n(n+1)`
C. `n(1+(1)/(n))^(2)`
D. None of these

Answer

A

Explanation

Let `S=1+2(1-(1)/(n))+3(1-(1)/(n))^(2)+"...."+oo`
`((1-(1)/(n))S=(1-(1)/(n))+2(1-(1)/(n))+"...."+oo)/(S(1-1+(1)/(n))=1(1-(1)/(n))+(1-(1)/(n))^(2)+"...."+oo)`
`implies (S)/(n)=(1)/(1-(1-(1)/(n))) " " [S_(oo)=(a)/(1-r) " by GP "]`
`implies S=(n)/((1)/(n))`
`implies S=n^(2)`.