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Good Approximations

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

There's a Limit

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?

Not Continued Fractions

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

Golden Fractions

Age 16 to 18
Challenge Level

Here is a sequence of continued fractions: $$X_1=1, \quad X_2 = {1 \over 1+1}, \quad X_3 = {1\over\displaystyle 1+ { 1 \over \displaystyle 1+ 1}}, \quad X_4 = {1\over\displaystyle 1+ { 1 \over \displaystyle 1+ { 1\over 1 + 1}}} ,...$$ Notice that $$X_{n+1} = {1\over 1 + X_n}.$$ Now suppose that this sequence tends to a limit $L$ as $n\to \infty$ then put $X_{n+1}=X_n=L$ and prove that $L =\phi - 1 = {1\over \phi}$ where $\phi$ is the Golden Ratio, the positive solution to the equation $x^2 - x -1 = 0$.

Prove that $$X_n={F_n\over F_{n+1}}$$ where $F_n$ is a Fibonacci number from the sequence defined by the relation $F_{n+2}=F_{n+1}+F_n$ where $F_1=1$ and $F_2=1$.

Hence show that the ratio of successive terms of the Fibonacci sequence $${F_{n+1}\over F_n}$$ tends to the Golden Ratio as $n\to \infty$.