For a given $A.P.$,$S_{20} = 100$ and $d = -2$. Then,$a = \ldots$

The $7^{th}$ and $12^{th}$ terms of an $A.P.$ are $46$ and $71$ respectively. Find the $n^{th}$ term of the $A.P.$

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

Find the sum of the multiples of $3$ lying between $250$ and $1000$.

An $AP$ consists of $37$ terms. The sum of the three middle most terms is $225$ and the sum of the last three is $429$. Find the $AP$.