CAT (QA) PYQs

Let $a_1, a_2, ...$ be integers such that $a_1 - a_2 + a_3 - a_4 + ... + (-1)^{n - 1} a_n = n$, for all $n \ge 1$. Then $a_{51} + a_{52} + ... + a_{1023}$ equals

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 a sequence of real numbers $x_1, x_2, ...... x_n$, if $x_1 - x_2 + x_3 - .... + (-1)^{n + 1} x_n = n^2 + 2n$ for all natural numbers n, then the sum $x_{49} + x_{50}$ equals