If `S_1.S_2.S_3` are the sum of the first n natural numbers, their squares and their cubes respectively, show that `9S_2^2=S_3(1+8S_1)`

If S_1, S_2, S_3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that 9 S_2^(2)= S_3(1+8 S_1)

Let S_(1) , S_(2), S_(3) , be the respective sums of first n, 2n and 3n. terms of the same arithmetic progression with a as the first term and d as the common difference. If R= S_(3)-S_(2)-S_(1) then R depends on

If S_(n) denotes the sum of first n terms of an A.P whose first term is a and (S_(nx))/(S_(x)) is independent of x then S_(p)=

If the sum of the first n terms of an Arithmetic progression is S_n=nX+1/2n(n-1)Y where X and Y are constants, find S_1 and S_2 .

If S_1,S_2 and S_3 are respectively the sums of n, 2n and 3n terms of an AP.Prove that S_3=3(S_2-S_1)

If S_1,S_2 and S_3 are respectively the sums of n, 2n and 3n terms of an A.P, whose first term is a and common difference d, then find S_1,S_2 and S_3

If S denotes the sum to infinity and S_(n) , denotes the sum of n terms of the series 1+(1)/(2)+(1)/(4) + (1)/(8)+ cdots , such that S-S_(n) lt (1)/(100) , then the least value of n is

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_(n) .

Let S_(1) be a square of side is 5 cm. Another square S_(2) is drawn by joining the mid-points of the sides of S_(1) . Square S_(3) is drawn by joining the mid-points of the sides S_(2) and so on. Then, (area of S_(1) + area of S_(2)+…+ area of S_(10) ) is a) 25(1-1/(2^(10)))cm^(2) b) 50(1-1/(2^(10)))cm^(2) c) 2(1-1/(2^(10)))cm^(2) d) 1-1/(2^(10))cm^(2)