In each of the following,$a$ and $d$ for an $A.P.$ are given. Find the $A.P.$ in each case. $a = -12, d = 3$.

Determine whether the following sequence is an $A.P.$ or not. (Assume that the pattern continues.) If it is an $A.P.$,find its $n^{th}$ term: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$

For the following $APs,$ write the first term and the common difference: $0.6, 1.7, 2.8, 3.9, \ldots$

Write the first three terms of the $APs$ when $a$ and $d$ are as given below:
$a=\sqrt{2}, d=\frac{1}{\sqrt{2}}$

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.