If the numbers $n-2, 4n-1$ and $5n+2$ are in $AP$,find the value of $n$.

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.

Find the sum of the first seven numbers which are multiples of $2$ as well as of $9$.

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.

The sum of three numbers in $A.P.$ is $12$ and the sum of their cubes is $288$. Find those numbers.

The $20^{th}$ term of the $A.P.$ $1, 11, 21, \ldots$ is........