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

If $X = \sum_{n=0}^\infty a^n$,$Y = \sum_{n=0}^\infty b^n$,and $Z = \sum_{n=0}^\infty c^n$,where $a, b, c$ are in arithmetic progression and $|a| < 1, |b| < 1, |c| < 1$,then $X, Y, Z$ are in:

If $(b+c), (c+a), (a+b)$ are in $H.P.$,then $a^2, b^2, c^2$ are in.......

If the $7^{th}$ term of a harmonic progression is $8$ and the $8^{th}$ term is $7$,then its $15^{th}$ term is

If $\frac{a_2 a_3}{a_1 a_4} = \frac{a_2 + a_3}{a_1 + a_4} = 3\left( \frac{a_2 - a_3}{a_1 - a_4} \right)$,then in which progression are $a_1, a_2, a_3, a_4$?

The numbers $log_{3}{2}, log_{6}{2}$ and $log_{12}{2}$ are in which progression?