**The Fibonacci sequence** is the sequence defined by:

First 15 sequence values:

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Fn | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 |

The number

$\phi =\frac{1+\sqrt{5}}{2}=1.61803...$

is called the Golden Ratio. It satisfies the property

$\phi =1+\frac{1}{\phi}$

Can you show that the sequence of ratio of successive Fibonacci numbers

$\frac{{F}_{n+1}}{{F}_{n}}$

converges to the Golden Ratio?

$\underset{n\to \infty}{\mathrm{lim}}\frac{{F}_{n+1}}{{F}_{n}}=\phi $

Observe that

${F}_{14}/{F}_{13}=377/233\approx 1.618025$

is already pretty close to φ.

What conditions must be met for this ratio to converge?

**Extension: **Missing Squares

An elegant variation of this idea makes use of two rectangles of such proportion that they fit side by side to make a perfect 8 by 8 checkerboard. When the pieces are rearranged to make the larger rectangle, there is an apparent gain in area of one square unit. Where did the extra square come from, and what does this have to do with Fibonacci?