In an $AP$,given $a=2, d=8, S_{n}=90$,find $n$ and $a_{n}$.

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?

Subba Rao started work in $1995$ at an annual salary of ₹ $5000$ and received an increment of ₹ $200$ each year. In which year did his income reach ₹ $7000$?

How many multiples of $4$ lie between $10$ and $250$?

$A$ sum of ₹ $1000$ is invested at $8 \%$ simple interest per year. Calculate the interest at the end of each year. Do these interests form an $AP$? If so,find the interest at the end of $30 \text{ years}$ making use of this fact. (in ₹)

If the sum of the first $n$ terms of an $AP$ is $4n - n^2$,what is the first term (that is $S_1$)? What is the sum of first two terms? What is the second term? Similarly,find the $3^{rd}$,the $10^{th}$ and the $n^{th}$ terms.