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

For the following $APs$,$-5, -1, 3, 7, \ldots$,write the first term and the common difference.

The houses of a row are numbered consecutively from $1$ to $49$. Show that there is a value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it. Find this value of $x$.

Are $2, 4, 8, 16, \ldots$ in an $AP$? If they form an $AP$,find the common difference $d$ and write three more terms.

$2, \frac{5}{2}, 3, \frac{7}{2}, \ldots$ are $APs$? If they form an $AP$,find the common difference $d$ and write three more terms.

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.