In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals

Find the sum of the following series up to n terms:

$6+.66+.666+\ldots$

If the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a $G.P.$ are $a, b$ and $c,$ respectively. Prove that

$a^{q-r} b^{r-p} c^{p-q}=1$

If the $4^{\text {th }}, 10^{\text {th }}$ and $16^{\text {th }}$ terms of a $G.P.$ are $x, y$ and $z,$ respectively. Prove that $x,$ $y, z$ are in $G.P.$

Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of its squares of first three terms is $33033$, then the sum of these three terms is equal to

The value of ${a^{{{\log }_b}x}}$, where $a = 0.2,\;b = \sqrt 5 ,\;x = \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + .........$to $\infty $ is