The Fibonacci numbers \(1, \ 1, \ 2,\ 3, \ 5,\ 8, 13,\ 21,\ 34, \ 55,\ 89,\ 144, \ \cdots\) are defined by the recurrence relation

\[F_1 = F_2 = 1, \ F_{n+2} = F_{n+1} + F_n \ \textrm{ for } n \ge 1.\]

Consider now the ratios \(\dfrac{F_{n+1}}{F_n}\):

\[\dfrac 11, \ \dfrac 21, \ \dfrac 32, \ \dfrac 53, \ \dfrac 85, \ \dfrac {13}8, \dfrac {21}{13}, \ \dfrac {21}{13}, \ \dfrac {34}{21}, \dfrac {55}{34}, \dfrac {89}{55}, \ \dfrac {144}{89}, \ \cdots \ .\]

The sequence of fractions appears to rise and fall alternately. This observation can be confirmed by reference to the equation

\[F_{n+1}^2 - F_nF_{n+2} = (-1)^n\]

which, when divided by \(F_{n+1}F_n\) gives

\[\dfrac{F_{n+1}}{F_n} - \dfrac {F_{n+2}}{F_{n+1}} = \dfrac {(-1)^n}{F_{n+1}F_n } \ ,\]

showing that the difference between successive terms in the sequence of ratios is alternately positive and negative.

Calculating the first few terms suggest that successive ratios converge to 1.618 … . This may be established by using Binet’s formula (see the previous post):

\[F_n = \dfrac 1{\sqrt 5}\Big(\Big(\dfrac{1+\sqrt 5}2 \Big)^n - \Big(\dfrac{1-\sqrt 5}2 \Big)^n\Big) \ \]

from which we have

\[\dfrac {F_{n+1}}{F_n} = \dfrac {\Big(\dfrac{1+\sqrt 5}2 \Big)^{n+1} - \Big(\dfrac{1-\sqrt 5}2 \Big)^{n+1}} {\Big(\dfrac{1+\sqrt 5}2 \Big)^n - \Big(\dfrac{1-\sqrt 5}2 \Big)^n}\]

Now look at the second terms in the numerator and denominator.

Since

\(-1< \dfrac{1-\sqrt 5}2 < 1\),

these terms tend to zero as \(n\) tends to infinity, so are insignificant in comparison with the first terms. It follows that

\[\frac {F_{n+1}}{F_n} \rightarrow \frac{1+\sqrt 5}2 \ \ - \ \ \textrm{the Golden Ratio}.\]

Check out the link between the golden spiral and the Fibonacci sequence here.

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Originally by John Webb