`S_n=sum n/(1+n^2+n^4) then S_10.S_20=` (A) `110.211/(421. 111) ` (B) `(110. 111) /(112. 421)` (C) `(110.420) /(421. 111)` (D) `(55.210)/(421. 111)`

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 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=1)^n(1+2+2^2+ .......+2^r)/(2^r), then S_n is equal to (a) 2^n n-1 (b) 1-1/(2^n) (c) n -1+1/(2^n) (d) 2^n-1

If S= tan^-1 (1/(n^2+n+1))+tan^-1 (1/(n^2+3n+3))+…+tan^-1 (1/(1+(n+19)(n+20))) then tan S is equal to (A) 20/(401+20n) (B) n/(n^2+20n+1) (C) n(401+20n) (D) 20/(n^2+n-1)

If sum_(n=1)^n u_n =an^2+bn+c, then |(u_1,u_2,u_3),(1,1,1),(7,8,9)|= (A) 0 (B) u_1-u_2+u_3 (C) 1 (D) none of these

If S_1,S_2,S_3,………….S_n denote the sum of 1,2,3…………n terms of an A.P. having first term a and S_(kx)/S_x is independent of x then S_1+S_2+S_3+……..+S_n= (A) (n(n+1)(2n+1)a)/6 (B) ^(n+2)C_3a (C) ^(n+1)C_3a (D) none of these

if S_n = C_0C_1 + C_1C_2 +...+ C_(n-1)C_n and S_(n+1)/S_n = 15/4 then n is

Let S_n=sum_(k=0)^n n/(n^2+kn+k^2) and T_n=sum_(k=0)^(n-1) n/(n^2+kn+k^2) for n=1,2,3... then (a) S_n lt pi/3sqrt(3) b) S_n gt pi/3sqrt(3) (c) T_n lt pi/3sqrt(3) (d) T_n gt pi/3sqrt(3)

If S_n=1^2+(1^2+2^2)+(1^2+2^2+3^2)+ upto n terms, then S_n=(n(n+1)^2(n+2))/(12) b. S_n=(n(n-1)^2(n+2))/(12) c. S_(22)=3276 d. S_(22)=23275

If S_n denotes the sum of n terms of A.P., then S_(n+3)-3S_(n+2)+3S_(n+1)-S_n= (a) S_2-n b. S_(n+1) c. 3S_n d. 0