b. The Fibonacci Series for Middle School StudentsThe 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)nWe 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 haveyleft  =  b(-1/a)nLet 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, 13Because 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