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.

(A) $(i)$ The fee charged from a student every month is $400, 400, 400, 400, \ldots$. This forms an $AP$ because the common difference $(d) = 400 - 400 = 0$ is constant.
$(ii)$ The fee charged for Classes $I$ to $XII$ is $250, (250+50), (250+2 \times 50), (250+3 \times 50), \ldots$,i.e.,$250, 300, 350, 400, \ldots$. This forms an $AP$ because the common difference $(d) = 300 - 250 = 50$ is constant.
$(iii)$ Simple interest for one year is $\frac{1000 \times 10 \times 1}{100} = 100$. The amount at the end of every year is $1000, (1000+100), (1000+200), (1000+300), \ldots$,i.e.,$1000, 1100, 1200, 1300, \ldots$. This forms an $AP$ because the common difference $(d) = 1100 - 1000 = 100$ is constant.
$(iv)$ Let the initial number of bacteria be $x$. Since they double every second,the sequence is $x, 2x, 4x, 8x, \ldots$. Here,$t_2 - t_1 = x$ and $t_3 - t_2 = 2x$. Since the difference is not constant,it does not form an $AP$.