The famous mathematician associated with finding the sum of the first $100$ natural numbers is

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

Find the sum of the integers between $100$ and $200$ that are not divisible by $9$.

The sum of three numbers in $A.P.$ is $-3$ and their product is $8$. Find those numbers.

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 fourth composite number in the $A.P.$ $5, 7, 9, 11, \ldots$ is......