The sum of the first and third term of an arithmetic progression is $12$ and the product of the first and second term is $24$. Find the first term.

If $a_1, a_2, \dots, a_n$ are positive real numbers such that their product is a constant $c$,then what is the minimum value of $a_1 + a_2 + \dots + a_{n-1} + 2a_n$?

$150$ workers were engaged to finish a job in a certain number of days. $4$ workers dropped out on the second day,$4$ more workers dropped out on the third day,and so on. It took $8$ more days to finish the work. Find the number of days in which the work was completed.

If the middle term of the $AP$ is $300$,then the sum of its first $51$ terms is

If three positive numbers $a, b,$ and $c$ are in $A.P.$ such that $abc = 8$,then the minimum possible value of $b$ is

The ratio of the sum of $m$ and $n$ terms of an $A.P.$ is $m^2:n^2$. Then the ratio of the $m^{th}$ and $n^{th}$ term will be: