Sequences and Series

Let ' $\boldsymbol{a}$ ' and ' $\mathbf{b}$ ' be two numbers where $\boldsymbol{a}<\boldsymbol{b}$. The geometric mean of these numbers exceeds the smaller number by 12 and the arithmetic mean is smaller than the larger number by 24 . Then ...

The product of three numbers in geometric progression is 8 and the sum of the product of the numbers taken in pairs is 14 . Find the numbers.

If $\boldsymbol{k}$ is the arithmetic mean of two given quantities and $\boldsymbol{p}, \boldsymbol{q}$ are the geometric means between the same two quantities, then $\boldsymbol{p}^{\mathbf{3}}+\boldsymbol{q}^{\mathbf{3}}$ is:

Every term of a geometric progression is positive, and every term is the sum of the two preceding terms. Then the common ratio of the geometric progression is:

The digits of a three-digit number taken in an order are in geometric progression. If one is added to the middle digit, they form an arithmetic progression. If 594 is subtracted from the number, then a new number with the same digits in rev...

$0.2+0.22+0.022+\ldots \ldots \ldots$. up to $n$ terms is equal to

A geometric progression consists of an even number of terms. If the sum of all the terms is five times the sum of the terms occupying the odd places, then the common ratio of the geometric progression is

Given that n number of arithmetic means are inserted between two pairs of numbers $a, 2 b$ and $2 a, b$; where $a, b \in R$. If the $m^{\text {th }}$ means in the two cases are the same, then the ratio $a: b$ is equal to

The terms of an infinitely decreasing geometric progression in which all the terms are positive, the first term is $\mathbf{4}$, and the difference between third and fifth term is $\frac{32}{81}$, then which of the following is not true

A geometric progression consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio of the G.P is

\((32) \times(32)^{\frac{1}{6}} \times(32)^{\frac{1}{36}} \times-----\infty \text { is equal to }\)

Given \(a, b, c\) are three unequal numbers such that \(\mathrm{b}\) is arithmetic mean of \(a\) and \(c\) and \((b-a),(c-b), a\) are in geometric progression. Then \(a: b: c\) is

If two positive numbers are in the ratio \(3+2 \sqrt{2}: 3-2 \sqrt{2}\), then the ratio between their A.M (arithmetic mean) and G.M (geometric mean) is

Consider an infinite geometric series with first term '\(a\)' and common ratio '\(r\)'.
If the sum of infinite geometric series is 4 and the second term is \(\frac{3}{4}\) then

A number consists of three digits in geometric progression. The sum of the right hand and left hand digits exceeds twice the middle digit by 1 and the sum of left hand and middle digits is two third of the sum of the middle and right hand d...

The sum of first three terms of a geometric progression is 16 and the sum of next three terms is 128 . The sum to \(\mathrm{n}\) terms of the geometric progression is

The sum of four numbers in a geometric progression is 60 , and the arithmetic mean of the first and the last number is 18 . Then the numbers are

\(\text { If } 6^{\text {th }} \text { term of a geometric progression is }-\frac{1}{32} \text { and } 9^{\text {th }} \text { term is } \frac{1}{256} \text { then } r \text { is }\)

Le \(x\) be the arithmetic mean and \(y, z\) be the two geometric means between any two positive numbers, then \(\frac{y^3+z^3}{x y z}=\) -----------

If three numbers \(a, b, c\) constitute both an A.P and G.P, then

If the sum of 12th and 22nd terms of an AP is 100, then the sum of the first 33 terms of an \(\mathrm{AP}\) is

The value of \(\frac{1}{2 !}+\frac{2}{3 !}+\ldots+\frac{99}{100 !}\) is equal to

The sum of \(n\) terms of the series, \(\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+\ldots\) is

Total number of elements in the power set of A containing 17 elements is

The first and fifth terms of an A.P. are \(-14\) and 2 respectively and the sum of its n terms is 40. The value of n is

If \(S = {{{2^2} - 1} \over 2} + {{{3^2} - 2} \over 6} + {{{4^2} - 3} \over {12}}\, + \,...\) upto 10 terms, then S is equal to

If a, b, c are in A.P., \(b-a,c-b\) and a are in G.P., then a : b : c is

The sum of the series \((1 + 2) + (1 + 2 + {2^2}) + (1 + 2 + {2^2} + {2^3}) + ....\) upto \(n\) terms is

\({1 \over {2\,.\,5}} + {1 \over {5\,.\,8}} + {1 \over {8\,.\,11}} + ............. + {1 \over {(3n - 1)(3n + 2)}} =\)