The sum of three consecutive terms of an $A.P.$ is $30$ and the product of its first and last term is $75$. Then,the common difference of the $A.P.$ is $\ldots \ldots \ldots \ldots .$

For an $A.P.$,$a=11$ and $d=7$. Find the sum of the first $40$ terms of the $A.P.$

$Rs. 280$ is to be distributed among four prizes. If the prize amount in each successive prize reduces by $Rs. 20$,find the amounts paid in those prizes.

For an $A.P.$,five times the $5^{th}$ term is equal to eight times the $8^{th}$ term. Find the $13^{th}$ term of the $A.P.$

Determine whether the following sequence is an $A.P.$ or not. (Assume that the pattern continues.) If it is an $A.P.$,find its $n^{th}$ term: $111, 107, 103, 99, \ldots$

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