If $a, x, y, z, b$ are in Arithmetic Progression ($A$.$P$.) such that $x + y + z = 15$,and if $a, x, y, z, b$ are in Harmonic Progression ($H$.$P$.) such that $1/x + 1/y + 1/z = 5/3$,find the values of $a$ and $b$.

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 the arithmetic mean of two positive numbers is $A$,their geometric mean is $G$,and their harmonic mean is $H$,then $H$ is equal to:

If $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $G.P.$ such that $a < b < c$ and $a+b+c = \frac{3}{4}$,then the value of $a$ is

Let $A_1, G_1, H_1$ be the arithmetic,geometric,and harmonic means of two distinct positive numbers. 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. Which of the following statements is true?

If the first and $(2n-1)$-th terms of an $AP$,$GP$,and $HP$ are equal and their $n$-th terms are respectively $a, b, c$,then always