Find the sum of the series: $7 + 10 \frac{1}{2} + 14 + \ldots + 84$.

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.

In each of the following,$a$ and $d$ for an $A.P.$ are given. Find the $A.P.$ in each case. $a = 21, d = -7$.

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 $20$ terms of the $A.P.$ $1, 21, 41, \ldots$ is:

Can any term of the $A.P.$ $14, 18, 22, \ldots$ be $142$? If yes,which term?