Let `a_1 , a_2 , ...., a_100` be an arithmetic progression with `a_1 = 3` and `S_p = sum_(j = 1)^p a_j , 1 le p le 100 . For any integer n with `1 le n le 20`, let m = 5n , if `S_m / S_n` does not n , then `a_2` is

Let a_1 , a_2 ,... be in harmonic progression with a_1 = 5 and a_20 = 25 . The least positive integer n for which a_n lt 0

If a_1,a_2,.....a_n are in H.P., then the expression a_1a_2 + a_2a_3 + ... + a_(n-1)a_n is equal to

Let a_1 , a_2 , a_3,.... be terms of an A.P. if (a_1 + a_2 +....+ a_p)/(a_1 + a_2 +...+ a_q) = p^2/q^2. p ne q then a_6/a_21 equals

Let a_1, a_2 , a_3,...., a_m be the arithmetic means between -2 and 1027 and let g_1, g_2, g_3.........,g_n be the geometric mean between 1 and 1024. g_1 g_2......g_n = 2^45 and a_1 + a_2 + a_3 +....+a_m = 1025 xx 171 The value of n is :

Let a_1, a_2 , a_3,...., a_m be the arithmetic means between -2 and 1027 and let g_1, g_2, g_3.........,g_n be the geometric mean between 1 and 1024. g_1 g_2......g_n = 2^45 and a_1 + a_2 + a_3 +....+a_m = 1025 xx 171 The value of m is :

Let (1+x^2)^2(1+x)^n=sum_(k=0)^(n+4)a_k x^k . If a_1 , a_2 and a_3 are in arithmetic progression, then the possible value/values of n is/are a. 5 b. 4 c. 3 d. 2

The number of ordered pairs of positive integers (m,n) satisfying m le 2n le 60 , n le 2m le 60 is

Suppose A_1 , A_2 ..... A_30 are thirty sets each with five elements and B_1,B_2....B_n are n sets each with three elements such that cap_(i=1)^30 A_i = cap_(j=1)^n B_j = S . If each elements of S belongs to exactly ten of the A_i s and exactly nine of the B_j's then the value of n is

Let a_1,a_2 ,a_3 ...... be an A.P. Prove that sum_(n=1)^(2m)(-1)^(n-1)a_n^2=m/(2m-1)(a_1^2-a_(2m)^2) .

If S_n denotes the sum to n terms of the series (1 le n le 9) 1 + 22 + 333 + .... + 999999999 then for n ge 2