Let $a_n = 1 + \frac{2}{n} - \frac{2}{n^3} - \frac{1}{n^4}$. For which smallest positive integer $n$ does the value of $P_n = a_2a_3a_4 \ldots a_n$ exceed $100$?
Source: Kyiv City MO 2021 Round 1, Problem 9.3
Tags: algebra
Let $a_n = 1 + \frac{2}{n} - \frac{2}{n^3} - \frac{1}{n^4}$. For which smallest positive integer $n$ does the value of $P_n = a_2a_3a_4 \ldots a_n$ exceed $100$?