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_(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_n denotes the sum of n terms of a G.P. whose first term and common ratio are a and r respectively, then : S_1 +S_3+S_5+.....+S_(2n-1)= (an)/(1-r)-(ar(1-r^(2n)))/((1-r)^2 (1+r)) .

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_(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

The sum of n, 2n, 3n terms of an A.P. are S_1, S_2, S_3 respectively. Prove that S_3 = 3(S_2 - S_1)

If S_(1), S_(2), S_(3),…., S_(n) are the sums to infinity of n infinte geometric series whose first terms are 1,2,3,… n and whose common ratios are (1)/(2), (1)/(3), (1)/(4), ….(1)/(n+1) respectively, show that, S_(1) + S_(2) + S_(3) +…S_(n) = (1)/(2)n(n+3)

If S_(n) denotes the sum fo n terms of a G.P. whose first term and common ratio are a and r respectively then show that: S_(1)+S_(3)+S_(5)+……..+S_(2n)-1=(an)/(1-r)-(ar(1-r^(2n)))/((1-r^(2))