In each of the following,$a$ and $d$ for an $A.P.$ are given. Find the $A.P.$ in each case. $a = \frac{15}{2}, \quad d = \frac{3}{2}$

Write the first four terms of the $AP,$ when the first term $a$ and the common difference $d$ are given as follows: $a = -2, d = 0.$

Verify that each of the following is an $AP$,and then write its next three terms.
$5, \frac{14}{3}, \frac{13}{3}, 4, \ldots$

The cost of digging a well after every metre of digging, when it costs ₹ $150$ for the first metre and rises by ₹ $50$ for each subsequent metre, does the list of numbers involved make an arithmetic progression, and why?

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 $x+1, 3x$ and $4x+2$ are three consecutive terms of an $A.P.$,find the value of $x$.