If $y = x - {x^2} + {x^3} - {x^4} + ......\infty $, then value of $x$ will be

If $a,\;b,\;c$ are ${p^{th}},\;{q^{th}}$ and ${r^{th}}$ terms of a $G.P.$, then ${\left( {\frac{c}{b}} \right)^p}{\left( {\frac{b}{a}} \right)^r}{\left( {\frac{a}{c}} \right)^q}$ is equal to

Let $a, a r, a r^2, \ldots . . .$. be an infinite $G.P.$ If $\sum_{n=0}^{\infty} a^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$, then $a+18 r$ is equal to :

Given $a_1,a_2,a_3.....$ form an increasing geometric progression with common ratio $r$ such that $log_8a_1 + log_8a_2 +.....+ log_8a_{12} = 2014,$ then the number of ordered pairs of integers $(a_1, r)$ is equal to

Given a $G.P.$ with $a=729$ and $7^{\text {th }}$ term $64,$ determine $S_{7}$

Let $a_1, a_2, a_3, \ldots .$. be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, \ldots .$. be a sequence of positive integers in geometric progression with common ratio $2$ . If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality

$2\left(a_1+a_2+\ldots .+a_n\right)=b_1+b_2+\ldots . .+b_n$

holds for some positive integer $n$, is. . . . . . .