Fibonacci Addition Trick

Date: 10/10/2002 at 23:29:56
From: Kate 
Subject: Fibonacci addition trick

When you write out 10 numbers in a Fibonacci sequence, multiplying the 
seventh number by 11 gives the total sum of all 10 numbers:

   2,3,5,8,13,21,34,55,89,144

   the sum is 374 and 34 x 11 = 374

I don't understand why this works for every Fibonacci sequence. Can 
you help me?

Date: 10/11/2002 at 02:44:39
From: Doctor Ian
Subject: Re: Fibonacci addition trick

Hi Kate,

That's a pretty interesting trick! Usually the way to explain 
something like this is to try it with variables, instead of with 
numbers. 

Suppose we start the sequence with the numbers a and b. Then the first 
10 terms are

  1.   a
  2.         b
  3.   a +   b
  4.   a +  2b
  5.  2a +  3b
  6.  3a +  5b
  7.  5a +  8b
  8.  8a + 13b
  9. 13a + 21b
 10. 21a + 34b

Now, what's the sum?  It's 

     55a + 88b

And what's the seventh term?  It's 

      5a + 8b

Kind of neat, isn't it? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/