Fibonacci numbers

The Fibonacci series follows this simple rule: You start out with 1 for the first two terms. Every term after that is the sum of the two terms before it. Mathematically, Fn = Fn-1 + Fn-2 – the sequence starts with 1, 1, 2, 3, 5, 8, 13, 21 and continues.

But why is this sequence special? First, it has a few patterns – every fifth number is divisible by 5, for example. Also, if you draw the diagonals on Pascal’s triangle, from left to right (bottom to top, of course), they’ll make the nth Fibonacci number (there’s a formula for it, but it looks too complicated to put here in text). If you divide a Fibonacci number by the one before it, it will be approximately equal to Φ (to be read as ‘phi’), which is ‘the golden ratio’.

Φ is basically defined as the solution to the equation x2 = x + 1, which, by the quadratic formula is equal to (5 + 1)/2. √5 ≈ 2.236, so Φ ≈ (3.236)/2, which means Φ ≈ 1.618… Note that Φ differs from π in that it can be proved that Φ is irrational, but π might be rational. Numbers like π and e are called transcendental.  

What significance does Φ have? It’s apparently the perfect proportion for a rectangle (or other shape) and it’s seen in architecture all the time – the Parthenon in Athens fits a rectangle with its sides in proportion to the golden ratio (length : width :: Φ : 1). Also, the Great Pyramid of Giza had a lateral surface height of 186.42 metres, and a distance from the edge to the centre of 115.2 metres. Guess what you get if you divide the two?

A rather interesting formula that uses Φ to calculate any Fibonacci number is as follows:

Fn = (Φn – (1 – Φ)n)/ √5

I mentioned earlier that every fifth Fibonacci number is a multiple of 5; similarly, every fourth Fibonacci number is a multiple of 3 (because F4 =3). Every third Fibonacci number is a multiple of 2; every tenth Fibonacci number is a multiple of 55, and so on.

Fibonacci numbers can be seen in nature as well; flowers usually have 3, 5, or 8 petals (less often 21, 34 or 55).

To finish, here’s a Fibonacci number trick: if you take any two consecutive Fibonacci numbers, then use them to find the next eight after that, the sum of all of these numbers is just the seventh number times 11! There’s a proof for it here.

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