If $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $H.P.$,then

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 $3.$

In a $G.P.$,the sum of three numbers is $14$. If $1$ is added to the first two numbers and subtracted from the third number,the series becomes an $A.P.$. Then,the greatest number is:

If the arithmetic mean of two numbers is $A$ and the geometric mean is $G$,then the numbers are

If the arithmetic mean of two numbers $a$ and $b$ is twice their geometric mean,then $a : b = \dots$

If $a, b, c$ are in $A.P.$,then $2^{ax + 1}, 2^{bx + 1}, 2^{cx + 1}$ for $x \ne 0$ are in