Home > Books, Equations & Formulae, Formulae, Foundation sets, GCSE Maths 2010, Higher sets, Maths blogging, Maths can be fun, Middle sets, Teaching ideas > Fibonacci Sequences – the Golden Rule about the Golden Ratio, which we just found out…

Fibonacci Sequences – the Golden Rule about the Golden Ratio, which we just found out…

We just found out something new….

It turns out that if you take the first ten terms of any Fibonacci sequence, the sum of those 10 terms is equal to the 7th term multiplied by 11.  Got it?

OK, let’s unpack that.  Here’s the most basic Fibonacci sequence:

1,1,2,3,5,8,13,21,34,55

If you plug these into your head/calculator, you will find that the sum of these 10 terms is 143.

Much quicker: take the 7th term (in this case 13), multiply that by 11 and you get, as if by magic… 143.

And it works for any Fibonacci sequence.  Let’s imagine one with bigger numbers starting with (to pick a number completely at random) 42:

42, 42, 84, 126, 210, 336, 546, 882, 1428, 2310

So we add all the terms up on a calculator and get 6,006.

Now, much quicker, jot down the 7th term (here: 546), multiply it by 11 and, hey presto: 546×11=6,006

It’s a trick you can cheerfully teach to your maths students – get them to take it home and impress their parents.  Something along the lines: “hey, dad/mum, I bet I can add up a Fibonacci sequence faster in my head than you can using a calculator”. 

Our thanks to the Republic of Maths blog that brought this to our attention, via this video link.

One final note: the covers on our (as in: Longman’s) AQA GCSE Maths Student Books this year were inspired by spirals.  A spiral galaxy on our Foundation sets book, a spiral staircase on our Higher sets book, and a chameleon’s tail on our Middle sets book (someone must’ve been wondering what this image was?) –

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