What a continued fraction is

Continued fractions are an alternative way of representing numbers: a rational number gives a finite continued fraction, an irrational one gives an infinite one. When an infinite continued fraction is periodic, the trick for computing its value is to exploit its self-repeating structure and reduce to a second-degree equation.

Example — Computing an infinite continued fraction

Let $f = 1 + \cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cdots}}}$ Notice that $\dfrac{5}{3} < f < 3$.

Trick: the repeating part equals the whole expression, hence: $f = 1 + \frac{2}{f} \implies f^2 = f + 2 \implies f^2-f-2=0 \implies \boxed{f = 2}$ (discarding $f=-1$ because $f>0$).

Links

Topics: Second-degree equations
Concepts: Second-degree equation · Continued fraction
Skills: Solving equations