The $n^{th}$ term of an $A.P.$ is given by $T_{n} = 2n - 1$. Then,the $10^{th}$ term of the $A.P.$ is.........

For a given $A.P.$,the $5^{th}$ term is $20$ and the $10^{th}$ term is $35$. Find the sum of the first $20$ terms of this $A.P.$

In an $AP$,if $S_{n} = 3n^{2} + 5n$ and $a_{k} = 164$,find the value of $k$.

$A$ person has to clear a debt of $Rs. 3250$ by monthly instalments. In the first month he has to pay $Rs. 20$ and then onwards every instalment is $Rs. 15$ more than the previous instalment. How long will it take to clear the debt?

Kanika was given her pocket money on Jan $1^{\text{st}}, 2008$. She puts $Rs.\, 1$ on Day $1$,$Rs.\, 2$ on Day $2$,$Rs.\, 3$ on Day $3$,and continued doing so till the end of the month,from this money into her piggy bank. She also spent $Rs.\, 204$ of her pocket money,and found that at the end of the month she still had $Rs.\, 100$ with her. How much was her pocket money for the month? (in $Rs.$)

Which of the following form an $AP$? Justify your answer.
$(i)$ $11, 22, 33, \ldots$
$(ii)$ $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$
$(iii)$ $2, 2^2, 2^3, 2^4, \ldots$
$(iv)$ $\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots$