The $n^{th}$ term of an $A.P.$ is denoted by:

For each of the following $A.P.s$,find the $n^{th}$ term: $\frac{4}{3}, 2, \frac{8}{3}, \frac{10}{3}, \ldots$

Divya deposited $Rs. 1000$ at compound interest at the rate of $10\%$ per annum. The amounts at the end of the first year,second year,third year,$\ldots,$ form an $AP$. Justify your answer.

The sum of the first $n$ terms of an $A.P.$ is given by $S_{n} = 3n^{2} + 5n$. 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.

If $S_{n}$ denotes the sum of the first $n$ terms of an $AP$,prove that $S_{12} = 3(S_{8} - S_{4})$.