A Wild Fraction

Let $a_n$ be the fraction where the digit $2$ appears $n$ times. Then:

$a_1 = \cfrac{1}{2}$

$a_2 = \cfrac{1}{2 - \frac{1}{2}} = \cfrac{2}{3}$

$a_3 = \cfrac{1}{2 - \frac{1}{2 - \frac{1}{2}}} = \cfrac{1}{2 - \frac{2}{3}} = \cfrac{3}{4}$

$a_4 = \cfrac{1}{2 - \frac{1}{2 - \frac{1}{2 - \frac{1}{2}}}} = \cfrac{1}{2 - \frac{3}{4}} = \cfrac{4}{5}$

$...$

or $a_n = \cfrac{1}{2 - a_{n - 1}}$ , which can be shown inductively to be $a_n = \cfrac{n}{n + 1}$ .

Therefore, $a_{2020} = \cfrac{2020}{2020 + 1} = \boxed{\cfrac{2020}{2021}}$ .