If $a, b, c$ are three distinct numbers in an arithmetic progression,and $b - a, c - b, a$ are in a geometric progression,then $a : b : c = .....$

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 $b$.

Find the value of $n$ so that $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ may be the geometric mean between $a$ and $b$.

If the $p^{th}, q^{th}, r^{th}$ and $s^{th}$ terms of an $A.P.$ are in $G.P.$,then $(p - q), (q - r), (r - s)$ will be in:

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

The $A.M., H.M.$ and $G.M.$ between two numbers are $\frac{144}{15}$,$15$ and $12$,but not necessarily in this order. Then $H.M., G.M.$ and $A.M.$ respectively are