The sum of the first two terms of a $G.P.$ is $36$ and the product of the first and the third terms is $9$ times the second term,then find the sum of the first $8$ terms.

If the harmonic mean between $a$ and $b$ is $H$,then the value of $\frac{1}{H - a} + \frac{1}{H - b}$ is

If the sum of three terms of a $G.P.$ is $19$ and their product is $216$,then the common ratio of the series is

Given $x = 1 + a + a^2 + ... \infty$ $(a < 1)$ and $y = 1 + b + b^2 + ... \infty$ $(b < 1)$,find the value of $1 + ab + a^2b^2 + ... \infty$.

If $a, b, c$ are in $A$.$P$.,then the straight line $ax + by + c = 0$ will always pass through the point

Let $S_1, S_2, \dots$ be squares such that for each $n \ge 1$,the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10 \text{ cm}$,then for which of the following values of $n$ is the area of $S_n$ less than $1 \text{ cm}^2$?