CAT (QA) PYQs

Consider a sequence of real number $x_1, x_2, x_3, ...$ such that $x_{n+1} = x_n + n - 1$ for all $n \ge 1$. If $x_1 = -1$ then $x_{100}$ is equal to

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 average of a non-decreasing sequence of $N$ numbers $a_{1}, a_{2}, \ldots, a_{N}$ is $300$. If $a_{1}$ is replaced by $6 a_{1}$, the new average becomes $400$. Then, the number of possible values of $a_{1}$ is