Find the $9^{th}$ term of the geometric progression $8 + 12 + 18 + 27 + \dots$

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \leq 0$,then:

Let the positive numbers $a_1, a_2, a_3, a_4$ and $a_5$ be in a $G$.$P$. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively,where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$,then $m+n$ is equal to $.........$.

If $a, b, c$ are in geometric progression,then which of the following is true?

If $a, b$ and $c$ are three distinct numbers in $G.P.$ and $a + b + c = xb$,then $x$ cannot be:

$0.14189189189...$ can be expressed as a rational number.