Let $x_{1}, x_{2}$ be the roots of $x^{2}-3x+a=0$ and $x_{3}, x_{4}$ be the roots of $x^{2}-12x+b=0$. If $x_{1} < x_{2} < x_{3} < x_{4}$ and $x_{1}, x_{2}, x_{3}, x_{4}$ are in $GP$, then $ab$ equals:

The geometric and harmonic means of two numbers $x_1$ and $x_2$ are $18$ and $16\frac{8}{13}$ respectively. The value of $|x_1 - x_2|$ is

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 $n$?

Let three real numbers $a, b, c$ be in arithmetic progression and $a+1, b, c+3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a, b$ and $c$ is $8$,then the cube of the geometric mean of $a, b$ and $c$ is

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 ratio of the harmonic mean to the geometric mean of two positive numbers $a$ and $b$ $(a > b)$ is $4 : 5$,then $a : b = \dots$ (in $: 1$)