The sum of $n$ arithmetic means between $a$ and $b$ is:

If $a_1, a_2, \dots, a_n$ are in $A.P.$ with common difference $d$,then the sum of the following series is $\sin d (\csc a_1 \csc a_2 + \csc a_2 \csc a_3 + \dots + \csc a_{n-1} \csc a_n)$

The sum of the first three terms of a $G.P.$ is to the sum of the first six terms as $125: 152$. Find the common ratio of the $G.P.$ (in $/5$)

$\prod\limits_{n = 1}^{10} {\left( {\frac{{\left( {6\sum\limits_{i = 0}^n i } \right) + 1}}{{\left( {6\sum\limits_{j = 0}^n {(j - 1)} } \right) + 1}}} \right)} $ is equal to:

If the roots of the equation $x^3 - 9x^2 + \alpha x - 15 = 0$ are in $A.P.$,then $\alpha$ is:

The product $(32) (32)^{1/6} (32)^{1/36} \ldots \infty$ is equal to