Sequences and Series
Then the number of elements in S, is :
12 + 2·22 + 32 + 2·42 + 52 + 2·62 + ....
If B – 2A = 100 λ, then λ is equal to :
A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224.
If the smaller of the numbers on the removed cards is k, then k − 20 =
Let be an arithmetic progression with and . For any integer n with 1, let m = 5n. If does not depend on n, then is
Let SK, k = 1,2…..100, denote the sum of the infinite geometric series whose first term is and the common ratio is . Then the value of
A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then
Suppose four distinct positive numbers are in G.P Let
Statement 1: The numbers are neither in A.P nor in G.P
Statement 2: The numbers are in H. P
If the total number of runs scored in n matches is , where n > 1, and the total number of runs scored in the kth match is , where 1 ≤ k ≤ n, what is the value of n?
Let a and b be positive real numbers. If a, A1, A2, b are in arithmetic progression; a, G1, G2, b are in geometric progression and a, H1, H2, b are in harmonic progression, then show that .
Let a1, a2, …………. be positive real numbers in geometric progression. For each n, let An, Gn, Hn be, respectively, the arithmetic mean, geometric mean, and harmonic mean of a1, a2, ………, an. Find an expression for the geometric mean of G1, G2, ………., Gn in terms of A1, A2, ………., An, H1, H2, ……….., Hn.