If $a_1, a_2, a_3, …….$ are in $A.P.$ such that $a_1 + a_7 + a_{16} = 40$, then the sum of the first $15$ terms of this $A.P.$ is

The ratio of the sums of $m$ and $n$ terms of an $A.P.$ is $m^{2}: n^{2} .$ Show that the ratio of $m^{ th }$ and $n^{ th }$ term is $(2 m-1):(2 n-1)$

Sum of the first $p, q$ and $r$ terms of an $A.P.$ are $a, b$ and $c,$ respectively. Prove that $\frac{a}{p}(q-r)+\frac{b}{q}(r-p)+\frac{c}{r}(p-q)=0$

The common difference of the $A.P.$ $b_{1}, b_{2}, \ldots,$ $b_{ m }$ is $2$ more than the common difference of $A.P.$ $a _{1}, a _{2}, \ldots, a _{ n } .$ If $a _{40}=-159, a _{100}=-399$ and $b _{100}= a _{70},$ then $b _{1}$ is equal to

If the sum of $n$ terms of an $A.P.$ is $3 n^{2}+5 n$ and its $m^{\text {th }}$ term is $164,$ find the value of $m$

If  ${\log _5}2,\,{\log _5}({2^x} - 3)$ and  ${\log _5}(\frac{{17}}{2} + {2^{x - 1}})$ are in $A.P.$ then the value of $x$ is :-