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# Comparing Continued Fractions

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Age 16 to 18

Challenge Level

Suppose $0 < a < b$. Which of the following continued fractions is bigger and why?

\[ \frac{1}{2+\frac{1}{3+\frac{1}{a}}} \] or \[ \frac{1}{2+\frac{1}{3+\frac{1}{b}}} \]

Suppose the fractions are continued in the same way, then which is the bigger in the following pair and why?

\[ \frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{a}}}} \]

or the same thing with b in place of a.

Now compare: $${1\over\displaystyle 2 + { 1 \over \displaystyle 3+ { 1\over \displaystyle 4 + \dots + {1\over\displaystyle 99+ {1\over \displaystyle {100 + {1 \over \displaystyle a}} }}}}}$$

and the same thing with $b$ in place of $a$.

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?