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# Golden Fractions

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 $n\to \infty$.

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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 $n\to \infty$.

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?