If the $A.M.$,$G.M.$,and $H.M.$ between two positive numbers $a$ and $b$ are equal,then

If $a, b, c$ are in $AP$,and $b-a, c-b, a$ are in $GP$,then $a: b: c$ is

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 altitudes of a triangle are in $A.P.$,then the sides of the triangle are in

If $a_1, a_2, \dots, a_n$ are positive real numbers such that $a_1 \cdot a_2 \cdot \dots \cdot a_n = 1$,then their sum is:

The geometric mean of two numbers is $6$ and their arithmetic mean is $6.5$. The numbers are