Let $a_1, a_2, a_3, \ldots, a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p=\sum_{i=1}^p a_i, 1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$, let $m=5 n$. If $\frac{S_{m m}}{S_n}$ does not depend on $n$, then $a_2$ is

What is the $20^{\text {th }}$ term of the sequence defined by

$a_{n}=(n-1)(2-n)(3+n) ?$

If ${S_1},\;{S_2},\;{S_3},………..{S_m}$ are the sums of $n$ terms of $m$ $A.P.'s$ whose first terms are $1,\;2,\;3,\;……………,m$ and common differences are $1,\;3,\;5,\;………..2m – 1$ respectively, then ${S_1} + {S_2} + {S_3} + …….{S_m} = $

Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots$ be an $A.P.$ If $\frac{a_{1}+a_{2}+\ldots+a_{10}}{a_{1}+a_{2}+\ldots+a_{p}}=\frac{100}{p^{2}}, p \neq 10$, then $\frac{a_{11}}{a_{10}}$ is equal to :

If $A$ be an arithmetic mean between two numbers and $S$ be the sum of $n$ arithmetic means between the same numbers, then