For a given $A.P.$,$a=2$ and $d=3$. Then,$S_{30} = \dots$

The ratio of the sums of first $n$ terms of two $A.P.s$ is $(7n + 1) : (4n + 27)$. Find the ratio of their $m^{th}$ terms.

The sum of the first $n$ terms of an $A.P.$ is given by $S_{n} = 3n^{2} + 5n$. Find the $n^{th}$ term of the $A.P.$

Find the sum:
$4-\frac{1}{n}+4-\frac{2}{n}+4-\frac{3}{n}+\ldots$ up to $n$ terms.

$A$ class has $49$ students who are allotted roll numbers from $1$ to $49$. Find a roll number $n$ such that the sum of all the roll numbers smaller than $n$ is equal to the sum of all the roll numbers greater than $n$.

Find the sum given below: $7 + 10 \frac{1}{2} + 14 + \ldots + 84$