Finking about Fibonacci

Driving home from Brighton today the circus / funfair was setting up in the park, and for some reason it reminded me of a trip to the funfair when I was little. I really can't remember very much about it, but I think that it was on the playing fields over the road from our house, and swingboats were almost certainly involved. What I do have a vague recollection of is coming home with prizes: a goldfish in a polythene bag and possibly a 7" vinyl record - how bizarre! Maybe my parents can help me with this and let me know if I am remembering correctly or if I am managing to piece together fragments of separate events.

Anyway, it was great to see Mum and Dad at the weekend at Bec's and entertaining to hear that Dad has already got the book that I have my eye on for holiday reading. As it seemed to get his seal of approval I have put my order in with Amazon and it should be arriving any day now. Apparently, one of the topics to look forward to in this popular maths tome deals with the Fibonacci sequence, a numerical sequence easy to grasp but with some fascinating properties. The first two numbers in the sequence are 0 and 1, and then new numbers are added onto the sequence by taking the sum of the two previous numbers. So, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on.

The particular property being considered is that if you take any number in the sequence and square it, then compare the answer with the product of the two numbers on either side of it in the sequence, the difference is always 1.

For example, 5 squared is 25, and 3 x 8 = 24 - difference 1. Similarly 8 squared is 64 and 5 x 13 = 65 - difference 1, and so on. Also worthy of note is that as you move along the sequence, the differences alternate between +1 and -1.

Being the fully paid up member of the amateur nerd society that I am, I have just bemused the rest of my family by writing up a proof about this and e-mailing it over to Dad!