CAT 2025 Slot 1 QA Question 15: For any natural number $k$, let $a_{k} = 3^{k}$. The smallest natural number $m$ for which
$\{(a_1)^1 imes (a_2)^2 imes \cdots imes (a_{20})^{20}\}

Question

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

Accepted Answer

Since $a_{k} = 3^{k}$, each term on the left contributes a power of 3.
$(a_i)^i = (3^i)^i = 3^{i^2}$
$	ext{Left exponent} = 1^2 + 2^2 + \cdots + 20^2$
$	ext{Left exponent} = \dfrac{20 	imes 21 	imes 41}{6} = 2870$

The right side is also a power of 3.
$	ext{Right exponent} = 21 + 22 + \cdots +