If the sum of the first $40$ terms of the series $3+4+8+9+13+14+18+19+\ldots$ is $(102)m$,then $m$ is equal to:

The sum of the first ten terms of an $A$.$P$. is $160$ and the sum of the first two terms of a $G$.$P$. is $8$. If the first term of the $A$.$P$. is equal to the common ratio of the $G$.$P$. and the first term of the $G$.$P$. is equal to the common difference of the $A$.$P$.,then the sum of all possible values of the first term of the $G$.$P$. is:

If $a, b, c$ are three consecutive terms of an $A.P.$ and $x, y, z$ are three consecutive terms of a $G.P.$,then the value of $x^{b-c} \cdot y^{c-a} \cdot z^{a-b}$ is:

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 $.....$

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:

For every natural number $n$,$n(n + 1)$ is always