# Leonardo of Pisa: Bunny Rabbits to Bull Markets - Patterns

Author(s):
Sandra Monteferrante

What is all the fuss about? Well, look closely at the Fibonacci numbers:

Terms f(1) f(2) f(3) f(4) f(5) f(6) f(7) f(8) f(9) f(10) f(11) f(12) f(13)
Values 1 1 2 3 5 8 13 21 34 55 89 144 233

Sums of selected Fibonacci numbers gives surprising answers. Add the 1st and 3rd elements (e.g. 1+2 = 3); you get the 4th! That is, in function notation, f(1) + f(3) = f(4). This is no aberration. By adding alternate elements we see a pattern:

 f(1) + f(3) + f(5) = f(6) 1+2+5=8 f(1) + f(3) + f(5) + f(7) = f(8) 1+2+5+13=21 f(1) + f(3) + f(5) + f(7) + f(9) = f(10) 1+2+5+13+34=55 f(1) + f(3) + f(5) + f(7) + f(9) + f(11) = f(12) 1+2+5+13+34+89=144

Adding all successive Fibonacci numbers gives another interesting pattern.

 f(1) + f(2) = f(4) - 1 1+1=3-1 f(1) + f(2) + f(3) = f(5) - 1 1+1+2=5-1 f(1) + f(2) + f(3) + f(4) = f(6) - 1 1+1+2+3=8-1 f(1) + f(2) + f(3) + f(4) + f(5)) = f(7) - 1 1+1+2+3+5=13-1

Actually, any number can be written as the sum of Fibonacci numbers. For example,
153 = 144 + 8 + 1 which is f(12) + f(6) + f(2).

Eduard Lucas was the first mathematician to study the Fibonacci numbers seriously for their mathematical properties. In fact, he made a general study of number sequences of this kind. They are called recursive sequences; that is, applying a fixed formula to previous terms generates new terms.

Not to be outdone, Lucas defined his own sequence, called, naturally, Lucas numbers. He began his sequence with 1 followed by 3 and then proceeded by adding terms the same way Fibonacci numbers are generated. Check them out below.

Terms l(1) l(2) l(3) l(4) l(5) l(6) l(7) l(8) l(9) l(10) l(11) l(12) l(13)
Values 1 3 4 7 11 18 29 47 76 123 199 322 512

If you look closely, you can see that Lucas numbers are, in fact, sums of Fibonacci numbers; that is l(2) = f(1) + f(3) (i.e. 3 = 1 + 2); l(3) = f(2) + f(4) (i.e. 4 = 1 + 3); l(4) = f(3) + f(5) (i.e. 7 = 2 + 5). This pattern continues; if we jump all the way to l(12), it is equal to f(11) + f(13) (i.e. 322 = 89 + 233). As you might expect, Lucas numbers and Fibonacci numbers are also linked with interesting connections.

Lesson Plan 3