b. The Fibonacci Series for Middle School Students

The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,    appears in the book The Da Vinci Code.  The author notes that the ratio of consecutive terms appears to approach a constant:
                                                            34 / 21   =  1.619 ...
                                                            55 / 34   =  1.6176 ...
                                                            89 / 55   =  1.618 ...
                                                             144 / 89 =  1.6179

That constant is approximately 1.618.  Thus, the series could be approximated by a function such as  yright  =  b(an),  where a = 1.618.

The series extends to the left as 13, -8, 5, -3, 2, -1, 1, 0.   Now we have to make use of some earlier learning.  A negative number times a negative number yields a positive number, so we must have an expression for this side, which is 

yleft  =  b(-a)n


We might guess whether n is positive or negative, or we could make use of the knowledge that a negative exponent implies division.  Then we would have

yleft  =  b(-1/a)n


Let us line up the arithmetic series  -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 along with the Fibonacci series so that the zero elements are aligned:

-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7
13, -8,  5, -3,  2,  -1,  1, 0, 1, 1, 2, 3, 5, 8, 13


Because a is greater than 1 (and 1/a is less than 1), as n becomes larger, (-1/a)n approaches zero, and as n becomes more negative, an becomes smaller.  To match the negative signs, the function for the whole series becomes y = b[(an) - (-1/a)n].  To calculate b, let us see what we get when n = 1:

y = 1 = b(1.618 + .618) = 2.236(b).   Therefore, b  =  1 / 2.236.


Thus, by pattern recognition we found the Fibonacci series to be described by the function  y  =  [(1.618n) - (-0.618)n] / 2.236.

We have used the engineering approach of applying the simple basics of negative numbers and exponents as we recognized patterns.

The same problem as seen by a high school student