If `t_n=sum_1^n n`, find `S_n=sum_1^n t_n`.

If t+n=sum_(1)^(n)n then find t_(n)^( prime)=sum_(1)^(n)t_(n)

Find sum_(n=1)^n u_n if u_n=sum_(n=0)^n1/2^n .

For the series sum t_(n,) S_n=sum_(n=1)^nt_n=2t_n-1 . Is the series geometric ? If so, find the sum to n terms of the series.

Let T_(n)=(1)/((sqrt(n)+sqrt(n+1))(4sqrt(n)+4sqrt(n+1))) and S_(n)=sum_(r=1)^(n)T_(r) then find S_(15) .

If sum of n termsof a sequende is S_n then its nth term t_n=S_n-S_(n-1) . This relation is vale for all ngt-1 provided S_0= 0. But if S_!=0 , then the relation is valid ony for nge2 and in hat cast t_1 can be obtained by the relation t_1=S_1. Also if nth term of a sequence t_1=S_n-S_(n-1) then sum of n term of the sequence can be obtained by putting n=1,2,3,.n and adding them. Thus sum_(n=1)^n t_n=S_n-S_0. if S_0=0, then sum_(n=1)^n t_n=S_n. On the basis of above information answer thefollowing questions:If nth term of a sequence is n/(1+n^2+n^4) then the sum of its first n terms is (A) (n^2+n)/(1+n+n^2) (B) (n^2-n)/(1+n+n^2) (C) (n^2+n)/(1-n+n^2) (D) (n^2+n)/(2(1+n+n^2)

Let n in N, S_n=sum_(r=0)^(3n)^(3n)C_r and T_n=sum_(r=0)^n^(3n)C_(3r), then |S_n-3T_n| equals

If sum of n terms of a sequence is S_n then its nth term t_n=S_n-S_(n-1) . This relation is valid for all ngt-1 provided S_0= 0. But if S_1=0 , then the relation is valid only for nge2 and in hat cast t_1 can be obtained by the relation t_1=S_1. Also if nth term of a sequence t_1=S_n-S_(n-1) then sum of n term of the sequence can be obtained by putting n=1,2,3,.n and adding them. Thus sum_(n=1)^n t_n=S_n-S_0. if S_0=0, then sum_(n=1)^n t_n=S_n. On the basis of above information answer the following questions: If the sum of n terms of a sequence is 10n^2+7n then the sequence is (A) an A.P. having common difference 20 (B) an A.P. having common difference 7 (C) an A.P. having common difference 27 (D) not an A.P.

Let S_n=sum_(r=0)^oo 1/n^r and sum_(n=1)^k (n-1)S_n = 5050 then k= (A) 50 (B) 505 (C) 100 (D) 55

If S_(n)=sum_(r=0)^(n)(1)/(nC_(r)) and sum_(r=0)^(n)(r)/(nC_(r)), then (t_(n))/(S_(n))=