How can you quickly calculate 2+3+5+8+13+21+34+55+89+144 before someone can calculate it with a calculator?

If you can work with the number 11, you can quickly calculate the sum of a Fibonacci sequence!  Below is a comic on how to quickly work with the number 11:

And more:

Now take a sequence of 10 numbers where you add the previous 2 to get the next… for example:

2+3+5+8+13+21+34+55+89+144=?

You can quickly calculate this by taking the 4th from last number and multiplying by 11 (in this case 34).  Use the comic from above to help you.  What did you get?  Does it add up?

There is one more gem to be found here (phi)… 144/89!

References:

S. Cram, Fun with phi from Grey Matters http://headinside.blogspot.com/2010/08/fun-with-phi.html Accessed 24 September 2012

A. Bogomolny, Fibonacci numbers trick from CTK Insights http://www.mathteacherctk.com/blog/2012/07/fibonacci-numbers-trick/ Accessed 24 September 2012

S. Strogatz, Proportion control from NYTimes.com http://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/ Accessed 24 September 2012