If `S_(n)` be the sum of first n terms of a G.P. whose common ratio is r, then show that,
`(r-1)(dS_(n))/(dr)=(n-1)S_(n)-nS_(n-1)`

Let S_(n) be the sum of first n terms of an A.P. If S_(2n) = 5s_(n) , then find the value of S_(3n) : S_(2n) .

Let S_n denote the sum of first n terms of an A.P. If S_(2n)=3S_n , then find the ratio S_(3n)//S_ndot

If S_(r ) be the sum of the cubes of first r natural numbers, then show that Sigma_(r=1)^(n) (2r+1)/(S_(r)) = (4n(n+2))/((n+1)^(2)) , for any natural number n.

If S_(n) be the sum of n consecutive terms of an A.P. show that, S_(n+3) - 3S_(n+2) + 3S_(n+1) - S_(n) = 0

If the sum of 1st n tenn of a GP is S_n=1 and S_(2n)=4 ,show that S_(3n):S_n=13:1

Show that , .^(n)C_(r)=(n-r+1)/(r).^(n)C_(r-1) .

If s_1 be the sum of (2n+1) terms of an A.P and s_2 be the sum its odd terms, then prove that s_1:s_2=(2n+1):(n+1)

If the sums of n, 2n and 3n terms of an A.P. be S_(1), S_(2), S_(3) respectively, then show that, S_(3) = 3(S_(2) - S_(1)) .

If an A.P., the sum of 1st n terms is 2n^2 + 3n , then the common difference will

If S be the sum of first (2n+1) terms of an A.P and the sum of terms in odd positions of these (2n+1) terms be S', then show that (n+1)S = (2n+1)S'.