If `P_n` is the sum of a `GdotPdot` upto `n` terms `(ngeq3),` then prove that `(1-r)(d P_n)/(d r)=(1-n)P_n+n P_(n-1),` where `r` is the common ratio of `GdotPdot`

Explanation

Let the first term of G.P. be `alpha`. Then
`P_(n)=alpha[(1-r^(n))/(1-r)]`
`(dp_(n))/(dr)=alpha[((1-r)(-nr^(n-1))+(1-r^(n)))/((1-r^())^(2))]`
`therefore" "(1-r)(dP_(n))/(dr)=alpha((-nr^(n-1)+nr^(n))/(1-r))+((1-r^(n))/(1-r))alpha`
`=alphan.((1.r^(n-1)-1+r^(n))/(1-r))+P_(n)`
`=ncdotP_(n-1)-nP_(n)+P_(n)`
`=(1-n)P_(n)+nP_(n-1)`