# Substitution (…and Continued Fractions)

This one’s really only for the Maths teachers of there. But I saw it and just had to reblog it! Awesome. SC.

Today in Precalculus I went on a bit of a 7 minute digression, talking about continued fractions. You see, a recursive problem showed up (we’re doing sequences): Write out the first five terms of the following sequence:

\$latex a_{n+1}=sqrt{2+a_n}\$ where \$latex a_1=sqrt{2}\$

So obviously they go like: \$latex a_1=sqrt{2}\$,\$latex a_2=sqrt{2+sqrt{2}}\$, \$latex a_3=sqrt{2+sqrt{2+sqrt{2}}}\$, \$latex a_4=sqrt{2+sqrt{2+sqrt{2+sqrt{2}}}}\$, and \$latex a_5=sqrt{2+sqrt{2+sqrt{2+sqrt{2+sqrt{2}}}}}\$

So great. Awesome. NOT. Booooring. So I showed them the decimal expansions:

\$latex approx 1.414, approx 1.848, approx 1.961, approx 1.990, approx 1.998, approx 1.999, approx 1.9998, approx 1.99996, approx 1.999991, approx 1.999997647\$

WHOA! This is getting closer and closer to 2… Weiiiird…

And then I say I can show them this will continue, and we can find a way to show that \$latex sqrt{2+sqrt{2+sqrt{2+sqrt{2+sqrt{2+…}}}}}\$ [where the pattern continues forever] will practically become 2.