If $A, G,$ and $H$ are the arithmetic mean,geometric mean,and harmonic mean of two given numbers respectively,then which of the following is true?

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an Arithmetic Progression are equal to the terms of a Geometric Progression,and these terms are $x, y, z$ respectively,then $x^{y - z} \cdot y^{z - x} \cdot z^{x - y} = \dots$

If $x, y, z$ are in Arithmetic Progression $(AP)$ and $x, y, t$ are in Geometric Progression $(GP)$,then in which progression are $x, x - y, t - z$?

If the sum of the arithmetic mean and the harmonic mean between two positive numbers is $25$ and their geometric mean is $12$,then what is the sum of the numbers?

Let $\frac{1}{16}, a$ and $b$ be in $G.P.$ and $\frac{1}{a}, \frac{1}{b}, 6$ be in $A.P.,$ where $a, b > 0$. Then $72(a + b)$ is equal to ...... .

If $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$ is the $A.M.$ of $a$ and $b$,then $n=$