How many terms of the $A.P.$ $5, 8, 11, \dots$ add up to $670$?

The sum of how many terms of the $A.P.$ $9, 17, 25, \ldots$ is $636$?

For an $A.P.$,$T_{n} = \ldots \ldots \ldots \ldots (n > 1)$

$A$ 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 $n$.

Kanika was given her pocket money on Jan $1^{\text{st}}, 2008$. She puts $Rs.\, 1$ on Day $1$,$Rs.\, 2$ on Day $2$,$Rs.\, 3$ on Day $3$,and continued doing so till the end of the month,from this money into her piggy bank. She also spent $Rs.\, 204$ of her pocket money,and found that at the end of the month she still had $Rs.\, 100$ with her. How much was her pocket money for the month? (in $Rs.$)

The $n^{th}$ term of an $A.P.$ is given by $T_{n} = 4n + 7$. Find the sum of $n$ terms of this $A.P.$