If $1, \log_9(3^{1-x} + 2), \log_3(4 \cdot 3^x - 1)$ are in $A.P.$, then $x$ equals:

The sum of $n$ terms of the following series $1 + (1 + x) + (1 + x + x^2) + \dots$ will be

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

The third term of a $G.P.$ is the square of the first term. If the second term is $8$,then the $6^{th}$ term is:

The first term of an $A.P.$ of consecutive integers is $p^2 + 1$. The sum of $(2p + 1)$ terms of this series can be expressed as:

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 $1 \text{ cm}^2$?