The harmonic mean of two numbers is $4$ and the arithmetic and geometric means satisfy the relation $2A + G^2 = 27$. The numbers are:

If $A.M.$ and $G.M.$ of two positive numbers $a$ and $b$ are $10$ and $8$ respectively,find the numbers.

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 $3.$

If the arithmetic mean and geometric mean of two positive numbers are $A$ and $G$ respectively,then the numbers are ..........

Suppose $a, b, c$ are in $A.P.$ and $a^{2}, 2b^{2}, c^{2}$ are in $G.P.$ If $a < b < c$ and $a+b+c=1,$ then $9(a^{2}+b^{2}+c^{2})$ is equal to . . . . . . .

If $a, b, c$ are in $A.P.$,then $10^{ax + 10}, 10^{bx + 10}, 10^{cx + 10}$ will be in