The four arithmetic means between $3$ and $23$ are

For a sequence $,a_1 = 2$ and $\frac{a_{n+1}}{a_n} = \frac{1}{3}.$ Then $\sum_{r=1}^{20} a_r $is

If $1+(1-2^{2} \cdot 1)+(1-4^{2} \cdot 3)+(1-6^{2} \cdot 5)+\ldots+(1-20^{2} \cdot 19) = \alpha - 220 \beta$,then an ordered pair $(\alpha, \beta)$ is equal to

If ${A_1}, {A_2}$; ${G_1}, {G_2}$ and ${H_1}, {H_2}$ are $AMs$,$GMs$,and $HMs$ between two quantities,then the value of $\frac{{G_1 G_2}}{{H_1 H_2}}$ is

If $|\alpha| < 1$ and $|\beta| < 1$,where $s_1 = 1 - \alpha + \alpha^2 - \alpha^3 + \dots \infty$ and $s_2 = 1 - \beta + \beta^2 - \beta^3 + \dots \infty$,then $1 - \alpha\beta + \alpha^2\beta^2 - \alpha^3\beta^3 + \dots \infty$ equals:

$A$ boy goes to school from his home at a speed of $x$ km/hour and comes back at a speed of $y$ km/hour,then the average speed is given by