If $A$,$G$,and $H$ are the arithmetic,geometric,and harmonic means between two positive real numbers,respectively,then:

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

If the first and $(2n - 1)^{th}$ terms of an $A.P.$,$G.P.$,and $H.P.$ are equal and their $n^{th}$ terms are respectively $a, b$ and $c$,then:

Let $b_i > 1$ for $i = 1, 2, \ldots, 101$. Suppose $\log _e b_1, \log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression $(A.P.)$ with the common difference $\log _e 2$. Suppose $a_1, a_2, \ldots, a_{101}$ are in $A.P.$ such that $a_1 = b_1$ and $a_{51} = b_{51}$. If $t = b_1 + b_2 + \cdots + b_{51}$ and $s = a_1 + a_2 + \cdots + a_{51}$,then:

If the arithmetic mean between $p$ and $q$ $(p > q)$ is twice the geometric mean,then $p : q = .......$

If $\alpha, \beta, \gamma$ are the geometric means between $ca, ab$; $ab, bc$; and $bc, ca$ respectively,where $a, b, c$ are in $A.P.$,then $\alpha^2, \beta^2, \gamma^2$ are in