Let $a_1=8, a_2, a_3, \ldots a_n$ be an $A.P.$ If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is

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

If $\log _{3} 2, \log _{3}\left(2^{x}-5\right), \log _{3}\left(2^{x}-\frac{7}{2}\right)$ are in an arithmetic progression, then the value of $x$ is equal to $.....$

The sums of $n$ terms of two arithmatic series are in the ratio $2n + 3:6n + 5$, then the ratio of their ${13^{th}}$ terms is

Let $a_{1}, a_{2} \ldots, a_{n}$ be a given $A.P.$ whose common difference is an integer and $S _{ n }= a _{1}+ a _{2}+\ldots+ a _{ n }$ If $a_{1}=1, a_{n}=300$ and $15 \leq n \leq 50,$ then the ordered pair $\left( S _{ n -4}, a _{ n -4}\right)$ is equal to

If the sum and product of the first three term in an $A.P$. are $33$ and $1155$, respectively, then a value of its $11^{th}$ tern is