Find the $25^{th}$ common term of the following $A.P.'s$

$S_1 = 1, 6, 11, .....$

$S_2 = 3, 7, 11, .....$

If the sum of $n$ terms of an $A.P$. is $2{n^2} + 5n$, then the ${n^{th}}$ term will be

Write the first five terms of the sequences whose $n^{t h}$ term is $a_{n}=n \frac{n^{2}+5}{4}$

Different $A.P.$'s are constructed with the first term $100$,the last term $199$,And integral common differences. The sum of the common differences of all such, $A.P$'s having at least $3$ terms and at most $33$ terms is.

If ${S_k}$ denotes the sum of first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively, then ${S_{kn}}/{S_n}$ be independent of $n$ if

The number of $5 -$tuples $(a, b, c, d, e)$ of positive integers such that

$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees

$II$. $a \leq b \leq c \leq d \leq e$

$III.$ $a, b, c, d, e$ are in arithmetic progression is