Let $S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+\ldots$. then $4 S$ is equal to

If $A=\sum_{n=1}^{\infty} \frac{1}{\left(3+(-1)^n\right)^n}$ and $B \sum_{n=1}^{\infty} \frac{(-1)^n}{\left(3+(-1)^n\right)^n}$, then $\frac{A}{B}$ is equal to :

Let $x, y>0$. If $x^3 y^2=2^{15}$, then the least value of $3 x+2 y$ is

Let $\left\{a_n\right\}_{n-0}^{\infty}$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=2 a_{n+1}-a_n+1$ for all $n \geq 0$. Then, $\sum_{n-2}^{\infty} \frac{a_n}{7^n}$ is equal to-

The sum $1+2 \cdot 3+3 \cdot 3^2+\ldots \ldots+10 \cdot 3^9$ is equal to

Let $\mathrm{A}=\left\{1, \mathrm{a}_1, \mathrm{a}_2 \ldots . . \mathrm{a}_{18}, 77\right\}$ be a set of integers with $1<\mathrm{a}_1<\mathrm{a}_2<\ldots<\mathrm{a}_{18}<77$. Let the set $\mathrm{A}+\mathrm{A}=\{\mathrm{x}+\mathrm{y}: \mathrm{x}$, $y \in A\}$ contain...

If the sum of the first ten terms of the series $\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+\ldots .$. is $\frac{m}{n}$, where $m$ and $n$ are co prime numbers, then $m+n$...

Let $\mathrm{A}=\sum_{i=1}^{10} \sum_{j=1}^{10} \min \{\mathrm{i}, \mathrm{j}\}$ and $\mathrm{B}=\sum_{i=1}^{10} \sum_{j=1}^{10} \max \{\mathrm{i}, \beta\}$. Then $\mathrm{A}+\mathrm{B}$ is equal to $\_\_\_\_$ .

Let $A_1, A_2, A_3, \ldots \ldots$ be an increasing geometric progression of positive real numbers. If $A_1 A_3 A_5 A_7=\frac{1}{1296}$ and $A_2+A_4=\frac{7}{36}$, then, the value...

The greatest integer less than or equal to the sum of first 100 terms of the sequence $\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \ldots$. is equal...