Find the sum of two-digit natural numbers divisible by $6$.

The sum of the first $n$ terms of an $A.P.$ is given by $S_{n} = 7n^{2} - 3n$. Find the $n^{th}$ term of the $A.P.$

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.

Which term of the $A.P.$ $85, 78, 71, \ldots$ is its first negative term? If the order of this term is $n$,find $S_{n}$.

Find whether $55$ is a term of the $AP: 7, 10, 13, \dots$ or not. If yes, find which term it is. (in $^{th}$)

How many terms of the $A.P.$ $5, 8, 11, \dots$ add up to $670$?