The number of common terms in the progressions $4, 9, 14, 19, \ldots$ up to $25^{\text{th}}$ term and $3, 6, 9, 12, \ldots$ up to $37^{\text{th}}$ term is:

If the integers from $1$ to $2021$ are written as a single integer like $123 \dots 91011 \dots 20202021$,then the $2021^{st}$ digit (counted from the left) in the resulting number is

Let $\{a_{n}\}_{n=0}^{\infty}$ be a sequence such that $a_{0}=0, a_{1}=0$ and $a_{n+2}=3a_{n+1}-2a_{n}+1$ for all $n \geq 0$. Then the value of $a_{25}a_{23}-2a_{25}a_{22}-2a_{23}a_{24}+4a_{22}a_{24}$ is equal to:

If $\frac{b + a}{b - a} = \frac{b + c}{b - c}$,then $a, b, c$ are in

If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that the $4^{\text{th}}$ $A.M.$ is equal to the $2^{\text{nd}}$ $G.M.$,then $m$ is equal to:

Let $x_{1}, x_{2}, x_{3}, \ldots, x_{20}$ be in a geometric progression with $x_{1} = 3$ and common ratio $r = \frac{1}{2}$. $A$ new data set is constructed by replacing each $x_{i}$ with $(x_{i} - i)^{2}$. If $\bar{x}$ is the mean of the new data,then the greatest integer less than or equal to $\bar{x}$ is $.....$