Theon's ladder to approximate the square root of two

Text

Theon’s ladder builds approximations of $\sqrt2$: from a fraction $\frac{a}{b}$ it moves to $\frac{a+2b}{a+b}$, which approximates better. Starting from $\frac{1}{1}$, apply the rule three times. Which fraction do you get? Compare it with $\sqrt2$.

Solution

Apply $\frac{a}{b}\mapsto\frac{a+2b}{a+b}$: $\frac{1}{1}\ \mapsto\ \frac{1+2}{1+1}=\frac{3}{2}\ \mapsto\ \frac{3+4}{3+2}=\frac{7}{5}\ \mapsto\ \frac{7+10}{7+5}=\frac{17}{12}.$ After three steps we get $\boxed{\dfrac{17}{12}}\approx 1.41667$, very close to $\sqrt2\approx 1.41421$.