If $a, b, c$ are in $A.P.$ and $a, c - b, b - a$ are in $G.P.$ $(a \ne b \ne c)$,then $a:b:c$ is

If the $2^{nd}, 5^{th},$ and $9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$,then the common ratio of this $G.P.$ is:

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

The ratio of the $A.M.$ and $G.M.$ of two positive numbers $a$ and $b$ is $m: n.$ Show that $a: b = (m + \sqrt{m^{2} - n^{2}}) : (m - \sqrt{m^{2} - n^{2}}).$

If $a^{1/x} = b^{1/y} = c^{1/z}$ and $a, b, c$ are in $G.P.$,then $x, y, z$ will be in

Let $A_1, G_1, H_1$ denote the arithmetic,geometric,and harmonic means,respectively,of two distinct positive numbers $a$ and $b$. For $n \geq 2$,let $A_n, G_n, H_n$ be the arithmetic,geometric,and harmonic means of $A_{n-1}$ and $H_{n-1}$ respectively.
$1.$ Which one of the following statements is correct?
$(A)$ $G_1 > G_2 > G_3 > \ldots$
$(B)$ $G_1 < G_2 < G_3 < \ldots$
$(C)$ $G_1 = G_2 = G_3 = \ldots$
$(D)$ $G_1 < G_3 < G_5 < \ldots$ and $G_2 > G_4 > G_6 > \ldots$
$2.$ Which of the following statements is correct?
$(A)$ $A_1 > A_2 > A_3 > \ldots$
$(B)$ $A_1 < A_2 < A_3 < \ldots$
$(C)$ $A_1 > A_3 > A_5 > \ldots$ and $A_2 < A_4 < A_6 < \ldots$
$(D)$ $A_1 < A_3 < A_5 < \ldots$ and $A_2 > A_4 > A_6 > \ldots$
$3.$ Which of the following statements is correct?
$(A)$ $H_1 > H_2 > H_3 > \ldots$
$(B)$ $H_1 < H_2 < H_3 < \ldots$
$(C)$ $H_1 > H_3 > H_5 > \ldots$ and $H_2 < H_4 < H_6 < \ldots$
$(D)$ $H_1 < H_3 < H_5 < \ldots$ and $H_2 > H_4 > H_6 > \ldots$
Give the answers for questions $1, 2,$ and $3.$