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 $x, y, z$ be three positive real numbers in a geometric progression such that $x<y<z$. If $5 x, 16 y$, and $12 z$ are in an arithmetic progression then the common ratio of the geometric progression is

If $a_1 + a_2 + a_3 + \dots + a_n = 3 \times (2^{n+1} - 2)$, for every $n \ge 1$, then $a_{11}$ equals