If $a_1, a_2, a_3, . . . , a_n, . . .$ are in $A.P.$ such that $a_4 - a_7 + a_{10} = m$,then the sum of the first $13$ terms of this $A.P.$ is .............. $m$.

If the sum of $n$ terms of an $A.P.$ is $3n^2 + 5n$ and $T_m = 164$,then $m = $

Let the sum of $n, 2n, 3n$ terms of an $A.P.$ be $S_{1}, S_{2}$ and $S_{3}$ respectively. Show that $S_{3} = 3(S_{2} - S_{1})$.

If $2x, x + 8$ and $3x + 1$ are in an arithmetic progression,then $x = \dots$

If the arithmetic mean between the $p^{th}$ term and $q^{th}$ term of an arithmetic progression is equal to the arithmetic mean between its $r^{th}$ and $s^{th}$ term,then $p + q = ......$

Four numbers are in arithmetic progression. The sum of the first and last term is $8$ and the product of both middle terms is $15$. The least number of the series is