In a geometric progression with $n$ terms and a common ratio of $3$,the sum of the $n$ terms is $364$ and the last term is $243$. Find the value of $n$.

Let ${a_n}$ be the ${n^{th}}$ term of a $G$.$P$. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha$ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta$,such that $\alpha \ne \beta$,then the common ratio is

Show that the products of the corresponding terms of the sequences $a, ar, ar^{2}, \dots, ar^{n-1}$ and $A, AR, AR^{2}, \dots, AR^{n-1}$ form a $G.P.$,and find the common ratio.

If $x = 1 + a + a^2 + \dots \infty$ $(a < 1)$ and $y = 1 + b + b^2 + \dots \infty$ $(b < 1)$,then the value of $1 + ab + a^2b^2 + \dots \infty$ is

If the product of three terms of a $G.P.$ is $512$. If $8$ is added to the first term and $6$ is added to the second term,the resulting numbers are in $A.P.$. Find the numbers.

The solution of the equation $1 + a + a^2 + a^3 + \dots + a^x = (1 + a)(1 + a^2)(1 + a^4)$ is given by $x$ is equal to