In an $AP$,if $S_{n} = n(4n + 1)$,find the $AP$.

Show that the sum of an $AP$ whose first term is $a,$ the second term is $b,$ and the last term is $c,$ is equal to $\frac{(a+c)(b+c-2a)}{2(b-a)}.$

The sequence $1, 3, 6, 10, \ldots$ is

For the finite $A.P.$ $40, 35, 30, \ldots, -200,$ find the $10^{th}$ term from the end.

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.

In the $AP: 10, 5, 0, -5, \ldots$,the common difference $d$ is equal to $5$. Justify whether the above statement is true or false.