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

If $\alpha, \beta, \gamma$ are the geometric means between $ca, ab$; $ab, bc$; and $bc, ca$ respectively,where $a, b, c$ are in $A.P.$,then $\alpha^2, \beta^2, \gamma^2$ are in

If the sum of the series $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + \dots + 2 \cdot (n-1)^2 + n^2$ (when $n$ is odd) or $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + \dots + 2 \cdot n^2$ (when $n$ is even) is given by $S_n = \frac{n(n+1)^2}{2}$ for even $n$,find the sum of the series when $n$ is odd.

If $a, b, c$ are in $A.P.$,then $\frac{(a - c)^2}{(b^2 - ac)} = $

If $x, y, z$ are in $A.P.$ and $\tan^{-1} x, \tan^{-1} y, \tan^{-1} z$ are also in another $A.P.$,then:

The sum of $24$ terms of the following series $\sqrt 2 + \sqrt 8 + \sqrt {18} + \sqrt {32} + \dots$ is