CAT (QA) PYQs

For some positive and distinct real numbers $x, y$ and $z$, if $\frac{1}{\sqrt{y}+\sqrt{z}}$ is the arithmetic mean of $\frac{1}{\sqrt{x}+\sqrt{z}}$ and $\frac{1}{\sqrt{x}+\sqrt{y}}$, then the relationship which will always hold true, is

The sum of the infinite series is $\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^{2}\left(\left(\frac{1}{5}\right)^{2}-\left(\frac{1}{7}\right)^{2}\right)+\left(\frac{1}{5}\right)^{3}\left(\left(\frac{1}{5}\right)^{3}-\left(\frac{1}{7}\right)^{3}\right)+\ldots .$. equal to

Suppose $x_{1}, x_{2}, x_{3}, \ldots, x_{100}$ are in arithmetic progression such that $x_{5}=-4$ and $2 x_{6}+2 x_{9}=x_{11}+x_{13}$, Then, $x_{100}$ equals