The ${20^{th}}$ term of the series $2 \times 4 + 4 \times 6 + 6 \times 8 + \dots$ will be

Let $a_1, a_2, a_3, ..., a_n$ be in an $A.P.$ If $a_3 + a_7 + a_{11} + a_{15} = 72$,then the sum of its first $17$ terms is equal to:

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms,the three terms now form an $A.P.$ Then the sum of the original three terms of the given $G.P.$ is

$1 + \frac{1^3 + 2^3}{1 + 2} + \frac{1^3 + 2^3 + 3^3}{1 + 2 + 3} + \dots + \frac{1^3 + 2^3 + 3^3 + \dots + 15^3}{1 + 2 + 3 + \dots + 15} - \frac{1}{2}(1 + 2 + 3 + \dots + 15)$ is equal to

Let $S_n$ and $s_n$ denote the sum of the first $n$ terms of two different $A.P.$ for which $\frac{s_n}{S_n} = \frac{3n - 13}{7n + 13}$. Find the ratio $\frac{s_n}{S_{2n}}$.

If the $A.M.$ and $G.M.$ of the roots of a quadratic equation are $8$ and $5$ respectively,then the quadratic equation will be