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

Let ${a_1}, {a_2}, \dots, {a_{30}}$ be an $A.P.$,$S = \sum_{i=1}^{30} {a_i}$ and $T = \sum_{i=1}^{15} {a_{2i-1}}$. If ${a_5} = 27$ and $S - 2T = 75$,then ${a_{10}}$ is equal to:

Consider the real-valued function $h: \{0, 1, 2, \ldots, 100\} \rightarrow \mathbb{R}$ such that $h(0) = 5$, $h(100) = 20$, and satisfying $h(p) = \frac{1}{2}\{h(p+1) + h(p-1)\}$ for every $p = 1, 2, \ldots, 99$. Then the value of $h(1)$ is:

In an $A.P.$,if the $p^{\text{th}}$ term is $\frac{1}{q}$ and the $q^{\text{th}}$ term is $\frac{1}{p}$,prove that the sum of the first $pq$ terms is $\frac{1}{2}(pq+1)$,where $p \neq q$.

If the $p^{th}$ term of an $A.P.$ is $q$ and the $q^{th}$ term is $p$,then its $r^{th}$ term will be:

Let $s_1, s_2, s_3, \ldots, s_{10}$ be the sum of the first $12$ terms of $10$ arithmetic progressions whose first terms are $1, 2, 3, \ldots, 10$ and whose common differences are $1, 3, 5, \ldots, 19$ respectively. Then $\sum_{i=1}^{10} s_i$ is equal to