Which term of the $A.P.$ $5, 15, 25, \ldots$ exceeds its $31^{st}$ term by $130$ (in $^{th}$)?

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 $a, b$ and $c$ such that the following numbers are in $AP : a, 7, b, 23, c$.

The production of a $TV$ factory increases equally every year. Its production in the $3^{rd}$ year was $600 \, TV$ and its production in the $7^{th}$ year was $700 \, TV$. Find the production in the $1^{st}$ year,in the $10^{th}$ year,and the total production in $7$ years.

Can any term of the $A.P.$ $14, 18, 22, \ldots$ be $142$? If yes,which term?

$3+6+9+12+\ldots+300 = \ldots$