If $y = x - x^2 + x^3 - x^4 + \dots \infty$,then the value of $x$ will be

If the sum of the $10$ terms of an $A.P.$ is $4$ times the sum of its $5$ terms,then the ratio of the first term to the common difference is:

If $a, b, c$ are in $H.P.$,then the value of $\left( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} \right) \left( \frac{1}{c} + \frac{1}{a} - \frac{1}{b} \right)$ is

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 $1 \text{ cm}^2$?

Find the common difference of an $A.P.$ whose first term is $100$ and the sum of whose first six terms is five times the sum of the next six terms.

If $(b+c), (c+a), (a+b)$ are in $H.P.$,then $a^2, b^2, c^2$ are in.......