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 the geometric mean between $a$ and $b$ is $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$,then what is the value of $n$?

If $A.M.$ of two terms is $9$ and $H.M.$ is $36$,then $G.M.$ will be

For two positive numbers $a$ and $b$,if $a, b$ and $\frac{1}{18}$ are in a geometric progression,while $\frac{1}{a}, 10$ and $\frac{1}{b}$ are in an arithmetic progression,then $16a + 12b$ is equal to $.........$.

If $A$ and $G$ are the $A.M.$ and $G.M.$ respectively between two positive numbers,prove that the numbers are $A \pm \sqrt{(A + G)(A - G)}$.

If $2(y - a)$ is the $H.M.$ between $y - x$ and $y - z$,then $x - a, y - a, z - a$ are in