Sequence of series practice · Medium

Suppose $a, b, c$ are distinct positive real numbers such that $a, 2 b, 3 c$ are in A.P. and $a, b, c$ are in G.P. The common ratio of G.P. is

If one geometric mean $G$ and two arithmetic means $p$ and $q$ are inserted between two numbers, then $G^2$ is equal to

The four numbers are in AP, such that their sum is 50 and greatest of them is 4 times the least, are

If 11 A.M.s are inserted between 28 and 10 , then the middle term in the series is

$3 \cdot \overline{537}=$

Three positive numbers form an increasing GP. If the middle terms in this GP is doubled, the new numbers are in AP. Then, the common ratio of the GP is

The common difference of an A.P. whose first term is unity and whose second, tenth and thirty fourth terms are in G.P., is

If the sum of $n$ terms of the series

$1 + \frac{4}{5} + \frac{7}{{{5^2}}} + \frac{{10}}{{{5^3}}} + \ldots$ is $l + \frac{{15}}{{16}}\left( {1 - \frac{1}{{{5^{n - 1}}}}} \right) - \frac{{\left( {3n - 2} \right)}}{{4\left( {{5^{n - 1}}} \right)}}$, then $l$ is

The sum of the series $8 !+\frac{9 !}{2 !}+\frac{10 !}{3 !}+\frac{11 !}{4 !}+\ldots$. up to 20 terms

If $a_1, a_2, a_3, \ldots, a_9$ are in G.P., then $\left|\begin{array}{lll}\log a_1 & \log a_2 & \log a_3 \\ \log a_4 & \log a_5 & \log a_6 \\ \log a_7 & \log a_8 & \log a_9\end{array}\right|=$

The maximum sum of an AP $40,38,36,34, \ldots$ is

If $\frac{{\log x}}{{{a^2} + ab + {b^2}}} = \frac{{\mathrm{log}y}}{{{b^2} + bc + {c^2}}} = \frac{{\log z}}{{{c^2} + ca + {a^2}}},$ then ${x^{a - b}}.{y^{b - c}}.{z^{c - a}} =$

$a, b, c, d$ are in GP and are in ascending order such that $a+d=112$ and $b+c=48$. If the GP is continued with $a$ as the first term, then the sum of the first six terms is

$\left\{ {{a_n}} \right\}\;and\;\left\{ {{b_n}} \right\}$ be two sequences given by ${a_n} = {\left( x \right)^{\frac{1}{{{2^n}}}}} + {\left( y \right)^{\frac{1}{{{2^n}}}}}$ and ${b_n} = {\left( x \right)^{\frac{1}{{{2^n}}}}} - {\left( y \right)^{\frac{1}{{{2^n}}}}}$ for all $n \in N,$ then ${a_1}{a_2}{a_3} \ldots {a_n}$ is equal to

Sum of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\ldots$ up to $n$ terms is equal to

If $1, \log _9\left(3^{1-x}+2\right), \log _3\left(4 \cdot 3^x-1\right)$ are in A.P., then $x=$

If $x=1+y+y^2+\ldots$, then $y$ is

In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49 , and the sum of the first and the third term is 35 . Then the first term of this geometric progression is:

If $a_{1}, a_{2}, \ldots, a_{n+1}$ are in AP, then $\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\cdots+\frac{1}{a_{n} a_{n+1}}$ is

If three numbers be in G.P., then their logarithms will be in

If ${\log _a}ab = x,$ then the value of ${\log _b}ab$ is

If the interior angles of a convex polygon are in A.P. with the smallest angle measuring 120 (in degrees) and the common difference between measures of angles being 5, then the polygon will have _____ sides.

If $(10)^{9}+2(11)^{1}\left(10^{8}\right)+3(11)^{2}(10)^{7}+\ldots$.
$+10(11)^{9}=k(10)^{9}$, then $k$ is equal to:

19 If $x$ be the $A M$ and $y, z$ be two GM's between two positive numbers, then $\frac{y^3+z^3}{x y z}$ is equal to-

If $a, b$ and $c$ are the sides of a $\triangle A B C$ are in AP, then $\cot \frac{C}{2}$ is equal to

If $S_n=\sum\limits_{{r=1}}^n \frac{1+2+2^2+\ldots \text { upto } r \text { terms }}{2^r}$, then $S_n$ is equal to

$n^{\text {th}}$ term of the series $2+4+7+11+\ldots$ will be

If the ratio of AM between two positive real numbers $a$ and $b$ to their HM is $m:n,$ then $a:b$ is equal to

If the sum of $n$ terms of a G.P. is ${S}$, their product is $p$ and the sum of their reciprocals is $R,$ then $p^2=$

If $a=\log _{12} 18$ and $b=\log _{24} 54$, then the value of $(a+b)^2+a(10-a)-b(10+b)$ is equal to