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+ 8S_1)`

The S.D for the set of the first 'n' natural numbers is

The sum of first n, 2n and 3n terms of an A.P. are S_(1), S_(2), S_(3) respectively. Prove that S_(3)=3(S_(2)-S_(1)) .

Let 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)) .

The sum of the squares of three distinct real numbers, which are in GP, is S^(2) . If their sum is a S , then show that a^(2) in ((1)/(3), 1) uu (1,3)

If S_(1), S_(2), S_(3),...,S_(n) are the sums of infinite geometric series, whose first terms are 1, 2, 3,.., n and whose common rations are (1)/(2), (1)/(3), (1)/(4),..., (1)/(n+1) respectively, then find the values of S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2) .

Sixteen players S_(1) , S_(2) , S_(3) ,…, S_(16) play in a tournament. Number of ways in which they can be grouped into eight pairs so that S_(1) and S_(2) are in different groups, is equal to

If S_(1), S_(2), S_(3), …, S_(m) are the sums of n terms of m A.P.'s whose first terms are 1, 2, 4, …, m and whose common differences are 1, 3, 5, …, (2m-1) repectively, then show that S_(1)+S_(2)+S_(3)+…+S_(n)=(1)/(2)mn(mn+1)

Let A B C be a triangle right-angled at Aa n dS be its circumcircle. Let S_1 be the circle touching the lines A B and A C and the circle S internally. Further, let S_2 be the circle touching the lines A B and A C produced and the circle S externally. If r_1 and r_2 are the radii of the circles S_1 and S_2 , respectively, show that r_1r_2=4 area ( A B C)dot