Find the sum of the first $17$ terms of an $AP$ whose $4^{\text{th}}$ and $9^{\text{th}}$ terms are $-15$ and $-30$ respectively.

The $26^{\text{th}}$,$11^{\text{th}}$,and the last term of an $AP$ are $0$,$3$,and $-\frac{1}{5}$,respectively. Find the common difference and the number of terms.

The $7^{th}$ term of an $A.P.$ is $108$ and the $11^{th}$ term is $212$. Then,its $n^{th}$ term is $\ldots \ldots \ldots \ldots$

Find the number of terms of the finite $A.P.$ $-1, -\frac{5}{6}, -\frac{2}{3}, \ldots, \frac{10}{3}$.

In which of the following situations,do the lists of numbers involved form an $AP$? Give reasons for your answers.
$(i)$ The fee charged from a student every month by a school for the whole session,when the monthly fee is $Rs. 400$.
$(ii)$ The fee charged every month by a school from Classes $I$ to $XII$,when the monthly fee for Class $I$ is $Rs. 250$,and it increases by $Rs. 50$ for the next higher class.
$(iii)$ The amount of money in the account of Varun at the end of every year when $Rs. 1000$ is deposited at simple interest of $10\%$ per annum.
$(iv)$ The number of bacteria in a certain food item after each second,when they double in every second.

The sum of the first $30$ positive multiples of $6$ is: