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

The sums of $n$ terms of three $A.P.'s$ whose first term is $1$ and common differences are $1, 2, 3$ are ${S_1}, {S_2}, {S_3}$ respectively. The true relation is

Three numbers form a $G.P.$ If the $3^{rd}$ term is decreased by $64$,the three numbers thus obtained will constitute an $A.P.$ If the second term of this $A.P.$ is decreased by $8$,a $G.P.$ will be formed again. Find the numbers.

The sum of $(n - 1)$ terms of the series $1 + (1 + 3) + (1 + 3 + 5) + \dots$ is

The $G.M.$ of the numbers $3, 3^2, 3^3, ..., 3^n$ is

The harmonic mean of $\frac{a}{1 - ab}$ and $\frac{a}{1 + ab}$ is