Three numbers are in $A.P.$ whose sum is $33$ and product is $792$. The smallest number among these is:

If $S_n$ denotes the sum of $n$ terms of an arithmetic progression,then the value of $(S_{2n} - S_n)$ is equal to

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

Let $f: R \rightarrow R$ be such that for all $x \in R$,the terms $(2^{1+x}+2^{1-x})$,$f(x)$,and $(3^x+3^{-x})$ are in $A.P.$. Then the minimum value of $f(x)$ is:

If the sum of the series $54 + 51 + 48 + \dots$ is $513$,then the number of terms is:

Given that $n$ $A$.$M$.'s are inserted between two sets of numbers $a, 2b$ and $2a, b$,where $a, b \in R$. Suppose further that the $m^{th}$ mean between these sets of numbers is the same,then the ratio $a:b$ equals