CAT (QA) PYQs

The average of all 3-digit terms in the arithmetic progression $38, 55, 72, \dots$, is

If n is a positive integer such that $(\sqrt[7]{10})(\sqrt[7]{10})^{2} \ldots(\sqrt[7]{10})^{n}>999$, then the smallest value of n is

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