The first term of a $G.P.$ is $1$. The sum of the third term and fifth term is $90$. Find the common ratio of the $G.P.$

If $a_{1} (>0), a_{2}, a_{3}, a_{4}, a_{5}$ are in a $G$.$P$.,$a_{2} + a_{4} = 2a_{3} + 1$ and $3a_{2} + a_{3} = 2a_{4}$,then $a_{2} + a_{4} + 2a_{5}$ is equal to

$\frac{6}{3^{26}} + \frac{10 \cdot 1}{3^{25}} + \frac{10 \cdot 2}{3^{24}} + \frac{10 \cdot 2^2}{3^{23}} + \ldots + \frac{10 \cdot 2^{24}}{3}$ is equal to

If $x = \sum\limits_{n = 0}^\infty {{a^n}} ,\;y = \sum\limits_{n = 0}^\infty {{b^n},\;z = \sum\limits_{n = 0}^\infty {{{(ab)}^n}} } $,where $a, b < 1$,then

The sum of the $3^{rd}$ and the $4^{th}$ terms of a $G.P.$ is $60$ and the product of its first three terms is $1000$. If the first term of this $G.P.$ is positive,then its $7^{th}$ term is

The $4^{th}$ term of a $G.P.$ is the square of its second term,and the first term is $-3$. Determine its $7^{th}$ term.