Here is a sequence of continued fractions:

X_{1} = 1, X_{2} = \frac{1}{1 + 1}, X_{3} = \frac{1}{1 + \frac{1}{1 + 1}}, X_{4} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + 1}}}, . . .

Notice that

X_{n + 1} = \frac{1}{1 + X_{n}} .

Now suppose that this sequence tends to a limit

L

n \to \infty

then put

X_{n + 1} = X_{n} = L

and prove that

L = ϕ - 1 = \frac{1}{ϕ}

where

ϕ

is the Golden Ratio, the positive solution to the equation

x^{2} - x - 1 = 0

Prove that

$X_{n} = \frac{F_{n}}{F_{n + 1}}$

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

$\frac{F_{n + 1}}{F_{n}}$

tends to the Golden Ratio as $n \to \infty$ .