If `S_k=(1+2+ ..........+k)/k ,` find the value of `S1^2+S2^2+.........+S n^2dot`

If S _(1), S _(2) , S _(3)……., S _(2n) are the sums of infinite geometric series whose first terms are respectively 1,2,3,…..,2n and common ratio are respectively, 1/2, 1/3, …….., (1)/(2n +1), find the value of , S_(1) ^(2) + S_(2) ^(2) +…....+ S _(2n -1) ^(2).

In an A.P. if S_1=T_1+T_2+T_3+.....+T_n (n is odd) S_2=T_2+T_4+T_6+.........+T_(n-1) , then find the value of S_1//S_2 in terms of ndot

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) .

In an AP, If S_(n)=3n^(2)+5n and a_(k)=164, then find the value of k.

In an AP, If S_(n)=3n^(2)+5n and a_(k)=164, then find the value of k.

S_(1),S_(2), S_(3),...,S_(n) are sums of n infinite geometric progressions. The first terms of these progressions are 1, 2^(2)-1, 2^(3)-1, ..., 2^(n) – 1 and the common ratios are (1)/(2),(1)/(2^(2)),(1)/(2^(3)), ...., (1)/(2^(n)) . Calculate the value of S_(1), +S_(2),+ ... + S_(n).

If for n in N ,sum_(k=0)^(2n)(-1)^k(^(2n)C_k)^2=A , then find the value of sum_(k=0)^(2n)(-1)^k(k-2n)(^(2n)C_k)^2dot

If S_(n) = 1 + (1)/(2) + (1)/(2^(2))+….+(1)/(2^(n-1)) , find the least value of n for which 2- S_(n) lt (1)/(100) .

Let S_(k) = (1+2+3+...+k)/(k) . If S_(1)^(2) + s_(2)^(2) +...+S_(10)^(2) = (5)/(12)A , then A is equal to

If S_n , be the sum of n terms of an A.P ; the value of S_n-2S_(n-1)+ S_(n-2) , is