Infinitely Nested Fractions with a Twist (Problem 1, Version 2)

Let $x = \dfrac{9}{9 + \dfrac{9}{9 + \ldots}} \implies x = \dfrac{9}{9 + x} \implies x^2 + 9x - 9 = 0 \implies x \approx 0.908$

$1000000000 + x \approx \boxed{1000000000.908}$

Let $x$ be the infinitely nested fraction. Then

$\begin{aligned} x & = \frac 9{9 + \frac 9{9 + \frac 9{9 + \ddots}}} \\ & = \frac 9{9+x} \\ 9x+x^2 & = 9 \\ x^2 + 9x - 9 & = 0 \\ \implies x & = \frac {-9+\sqrt{81+36}}2 \approx 0.908 & \small \blue{\text{Since }x > 0} \end{aligned}$

Since the infinitely nested fraction has $1,000,000,000$ on the side, the answer is $1,000,000,000$ as shown below: But when you put $0$ on the side: Therefore, the true answer is $1,000,000,000.9083269132$