part of a talk for sixth-formers

Each Fibonacci Number (FN) is the sum of its two immediate predecessors, starting with 0 and then 1. So the sequence is 0,1,1,2,3,5,8,13,21,34,55,89,....
A practical application is a quick and rough conversion between miles (m) and kilometers (k) using successive FNs.

That is, after the first few, we estimate 2m=3k, 3m=5k, 5m=8k, 8m=13k, 13m=21k, .... and so on, good to the nearest whole number.

Evidently neighbouring FNs settle to a fixed ratio of around 1.6, which by chance is just about the mile-kilometer conversion factor.

How does this work?

Well, given a FN, if the next one is a factor x times bigger then the next-but-one is two such factors bigger. From this, the summing rule gives x² = x + 1.

This is a quadratic equation for the value of x. Remembering what you learned at school, you soon find that x = 1.61809887... is the relevant solution.

In fact of course 1 mile = 1.609344 kilometer, but the FNs are not too far away for everyday round-number purposes.

Notice that the value for x depends only on the summing rule and not on the two values 0,1 that start the FN sequence.

From 2 and 1 for instance, you get the Lucas Numbers 2,1,3,4,7,11,18,28,.... So, after they settle down, there are similar approximations 4m=7k, 7m=11k, etc.

Likewise, every marathon runner knows that 26m=42k, and so by adding you get 42m=68k. Thus a British driver faced with a French speed limit of 70kph must not exceed about 43mph.

... stuff about Binet left out ...
As it happens, the number 1.618039887... that comes from the Fibonacci sequence was already significant to the Greeks, 1700 years or so before our hero.

Evidently Pythagoras and others thought that the most aesthetically pleasing way to divide a length in two was such that the ratio of the whole distance (a+b) to its larger part (a) is equal to the ratio of that larger part (a) to the smaller part (b).

This means that (a+b)/a = a/b.

From this you can easily deduce (can't you?) that therefore (a/b)² = (a/b) + 1.

Then notice the coincidence — this is the same equation as for x above!

Thus a/b = 1.618039887.... too, and so if a+b = 1 then a = 0.618039887....

That is, the Pythagoras division is roughly 62% to 38%.

Nowadays this "division in divine proportion" is called a "golden section" and the ratio 62:38 (or the number 1.61809887... itself) is called the Golden Ratio (GR).

Down the centuries, sympathetic eyes have seen the GR all over the place in art, architecture and music, as well as in nature.

But for example even though some ancient monument-builders had the means to construct a practical golden section, there's no evidence they actually did.

While many GR claims are misconception and flim-flam, some are hardly surprising.

For instance the occurrence of FNs and the GR in the structure of plants is well understood from biology and dynamics.

And the artist's empirical Rule of Thirds for composition advocates a division of about 67:33. For placement of sizable objects this isn't much different from 62:38. A painting's rule-of-thirds layout may easily be mistaken for a GR construction.

Unfortunately, riding on the coat-tails of aspects of art and nature, there's also a whole industry of delusion over the stock market. This is more than just harmless wishful thinking — it can hit your wallet. Beware of snake-oil, coloured golden and flavoured with Fibonacci!