[" If "S_(1),S_(2),...,S_(n)" are the sums of "n" terms of "n" G.P.'s whose first term is "1" in each and common "],[" ratios are "1,2,3,...,n" respectively,then prove that "]

If S_(1),S_(2),...,S_(n) are the sum of n term of n G.P.whose first term is 1 in each and common ratios are 1,2,3,...,n respectively, then prove that S_(1)+S_(2)+2S_(3)+3S_(4)+...(n-1)S_(n)=1^(n)+2^(n)+3^(n)+...+n^(n)

If S_1, S_2, ,S_n are the sum of n term of n G.P., whose first term is 1 in each and common ratios are 1,2,3, ,n respectively, then prove that S_1+S_2+2S_3+3S_4+(n-1)S_n=1^n+2^n+3^n++n^ndot

If S_1,S_2,S_3,...,S_n be the sums of first n terms of n G.P.'s whose first terms are each unity and the common ratios are 1, 2, 3,.....,n respectively, prove that S_1+S_2+2S_3+3S_4+.........+(n-1)S_n=1^n+2^n+3^n+....+n^n.

If s_(1), s_(2), s_(3)…..s_(p) are the sums of n terms of 'p' A.P's whose first terms are 1,2,3…p and common differences are 1,3,5… (2p-1) respectively then s_(1) + s_(2) + s_(3) + …+ s_(p) =

If S_(n) represents the sum of n terms of G.P whose first term and common ratio are a and r respectively, then s_(1) + s_(2) + s_(3) + …+ s_(n) =

If S_(1), S_(2), S_(3),….., S_(n) are the sum of infinite geometric series whose first terms are 1,3,5…., (2n-1) and whose common rations are 2/3, 2/5,…., (2)/(2n +1) respectively, then {(1)/(S_(1) S_(2)S_(3))+ (1)/(S_(2) S_(3) S_(4))+ (1)/(S_(3) S_(4)S_(5))+ ........."upon infinite terms"}=

if S_(n) denotes the sum of n terms of a G.P whose first term is a and common ratio is r then find the sum of S_(1),S_(3),S_(5).........,S_(2)n-1

Prove that the sum of n terms of GP with first term a and common ratio r is given by S_(n)=a(r^(n)-1)/(r-1)