CAT (QA) PYQs

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

The natural numbers are divided into groups as (1), $(2,3,4),(5,6,7,8,9), \ldots .$. and so on. Then, the sum of the numbers in the 15th group is equal to