CAT (QA) PYQs

For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which $\{(a_1)^1 \times (a_2)^2 \times \ldots \times (a_{20})^{20}\} < \{a_{21} \times a_{22} \times \ldots \times a_{(20+m)}\}$, is

Let $a_{1}, a_{2}, \ldots \ldots . . . a_{3 n}$ be an arithmetic progression with $a_{1}=3$ and $a_{2}=7$. If $a_{1}+a_{2}+\ldots .+a_{3 n}=1830$, then what is the smallest positive integer $m$ such that $m\left(a_{1}+a_{2}+\ldots .+a_{n}\right)>1830$ ?

The sum of the infinite series is $\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^{2}\left(\left(\frac{1}{5}\right)^{2}-\left(\frac{1}{7}\right)^{2}\right)+\left(\frac{1}{5}\right)^{3}\left(\left(\frac{1}{5}\right)^{3}-\left(\frac{1}{7}\right)^{3}\right)+\ldots .$. equal to