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...

If $x=\sum_{n=0}^{\infty} a^n, y=\sum_{n=0}^{\infty} b^n, z=\sum_{n=0}^{\infty} c^n$, where $a, b, c$ are in A.P. and $|a|<1,|b|<1,|c|<1 \neq 0$, then

If $\left\{a_i\right\}_{i=1}^n$ where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum_{i=1}^n a_i=192$, $\sum_{i=1}^{n / 2} a_{2 i}=120$,...

Suppose $a_1, a_2, 2, a_3, a_4$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum...

Let $[\alpha]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[\sqrt{120}]$ is equal to $\_\_\_\_$ .

If $\operatorname{gcd}(m, n)=1$ and $1^2-2^2+3^2-4^2+\ldots \ldots+(2021)^2-(2022)^2+(2023)^2=1012 m^2 n$, then $m^2-n^2$ is equal to