If $a, b, c$ are in a geometric progression and $a^x = b^y = c^z$,then in which progression are $x, y, z$?

Let $a, b, c, p, q$ and $r$ be positive real numbers such that $a, b$ and $c$ are in $GP$ and $a^{p} = b^{q} = c^{r}$. Then,

If $\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}$ is the harmonic mean between $a$ and $b$,then the value of $n$ is

Five numbers are in $HP$. The middle term is $1$ and the ratio of the second and the fourth terms is $2:1$. Then, the sum of the first three terms is

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 . . . . progression.

$A$ boy goes from his home to school at a speed of $x \text{ km/hr}$ and returns at a speed of $y \text{ km/hr}$. What is the average speed of the boy?