Sequences and Series

If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

Let a₁, a₂, . . . . . ., a₁₀ be a G.P.    If ${{{a_3}} \over {{a_1}}} = 25,$ then ${{{a_9}} \over {{a_5}}}$ equals

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is ${{27} \over {19}}$.Then the common ratio of this series is :

Let a, b and c be the 7^th, 11^th and 13^th terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then ${a \over c}$ equal to :

The sum of series ${1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......$ upto infinity is

Let $a$, b and c be in G.P. with common ratio r, where $a$ $\ne$ 0 and 0 < r $\le$ ${1 \over 2}$ . If 3$a$, 7b and 15c are the first three terms of an A.P., then the 4^th term of this A.P. is :

If m is the A.M. of two distinct real numbers l and n $(l,n > 1)$ and ${G_1},{G_2}$ and ${G_3}$ are three geometric means between $l$ and n, then $G_1^4\, + 2G_2^4\, + G_3^4$ equals:

Let x, y, z be positive real numbers such that x + y + z = 12 and x³y⁴z⁵ = (0.1) (600)³. Then x³ + y³ + z³is equal to :

If   A > 0, B > 0   and    A + B = ${\pi \over 6}$, then the minimum value of tanA + tanB is :

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then ${{a + b} \over {a - b}}$ is equal to :

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression ${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}$ is :

If sin⁴$\alpha$ + 4 cos⁴$\beta$ + 2 = 4$\sqrt 2$ sin $\alpha$ cos $\beta$; $\alpha$, $\beta$ $\in$ [0, $\pi$], then cos($\alpha$ + $\beta$) $-$ cos($\alpha$ $-$ $\beta$) is equal to :

The minimum value of 2^sinx + 2^cosx is :

The minimum value of $f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$, where a, $x \in R$ and a > 0, is equal to :

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

Let x, y > 0. If x³y² = 2¹⁵, then the least value of 3x + 2y is

Consider two G.Ps. 2, 2², 2³, ..... and 4, 4², 4³, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is ${(2)^{{{225} \over 8}}}$, then $\sum\limits_{k = 1}^n {k(n - k)}$ is equal to :

Let a₁, a₂, a₃, . . . . . . . , a_n, . . . . . be in A.P. If a₃ + a₇ + a₁₁ + a₁₅ = 72, then the sum of its first 17 terms is equal to :