CAT (QA) PYQs

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is $(n + 2n^2)$. If the $n^{th}$ term of the progression is divisible by $9$, then the smallest possible value of $n$ is

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