Answer:
2020
Step-by-step explanation:
This is an example of what is called an "infinite continued fraction".
For the repeated fraction:
y = 2019 / (2 + 2019 / (2 + 2019 / (2 + 2019 / ...
We can rewrite using substitution:
y = 2019 / (2 + y)
Solving for y:
y (2 + y) = 2019
2y + y² = 2019
y² + 2y + 1 = 2020
(y + 1)² = 2020
y + 1 = ±√2020
y = -1 ± √2020
Since the repeated fraction must be positive, y = -1 + √2020.
Therefore:
x = (1 + y)²
x = (1 + -1 + √2020)²
x = 2020
