how to geometrically explain why pell numbers close to sqrt 2

Question

How to graphically explain that the limit of yn/xn = $\sqrt 2$, as n approaches to infinity? Like, I know how to prove it algebraically.

Answer 1

Perhaps not quite what you want, but an explanation how the heron-method works.

We want to determine the length of a square with area $2$. We start with the rectangular with sides $1$ and $2$. To get closer to a square, we take the arithmetic mean of the sides (which is $\frac{3}{2}$ and determine the corresponding side (which is $\frac{4}{3}$). This can be repeated and leads to the iteration $$x_{n+1}=\frac{x_n+\frac{2}{x_n}}{2}=\frac{x_n^2+2}{2x_n}$$ which is exactly the heron-method. If we start with $x_0=1$, the fractions are among the partial fractions resulting from the continued fraction.