If `sum_(n=1)^k [1/3+n/90]=21` where [x] dentes the integeral part of x, then k= (A) 84 (B) 80 (C) 85 (D) none of these

sum_(k=1)^n k^3=

If f:(0,pi)rarr R and is given by f(x)=sum_(k=1)^(n)[1+sin((kx)/(n))] where [] denotes the integral part of x, then the range of f(x) is

If a_1,a_2,a_3 are in G.P. having common ratio r such that sum_(k=1)^n a_(2k-1)= sum_(k=1)^na_(2k+2)!=0 then number of possible value of r is (A) 1 (B) 2 (C) 3 (D) none of these

The period of f(x)=[x]+[2x]+[3x]+[4x]+...[nx]-(n(n+1))/(2)x where n in N, is (where [.] represents greatest integer function).n (b) 1 (c) (1)/(n) (d) none of these

If a,b,c are proper fractiion are in H.P. and x sum_(n=1)^oo a^n, y=sum_(n=1)^oo b^n, z= sum_(n=1)^oo c^n then x,y,z are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

If sum_(r=1)^20(r^2+1)r!=k!20 then sum of all divisors of k of the from 7^n, n epsilin N is (A) 7 (B) 58 (C) 350 (D) none of these

If (2+x)(2+x^2)(2+x^3)………..(2+x^100)=sum_(r=0)^n k_rx^r, then n= (A) 2550 (B) 5050 (C) 2^8 (D) none of these

The coefficient of x^k d in the expansion of 1+(1+x)+(1+x_)^2+……+(1+x)^n is (A) ^nC_k (B) ^(n+1)C_k (C) ^(n+1)C_(k+1) (D) none of these

Lt_(nrarroo) {(n!)/(kn)^n}^(1/n), k!=0 , is equal to (A) k/e (B) e/k (C) 1/(ke) (D) none of these