In the preceding chapter, we learned how Fibonacci numbers can be generated from Pascal's triangle. We now turn our attention to how Fibonacci and Lucas numbers can be computed from similar triangular arrays that have Pascal-like features.
In 1966, N.A. Draim of Ventura, California, and M. Bicknell of A.C. Wilcox High School, Santa Clara, California, studied the sums and differences of like-powers of the solutions of the quadratic equation . They were also studied in 1977 by J.E. Woko of Abia State Polytechnic, Aba, Nigeria . As we will see shortly, an intriguing relationship exists between these expressions, and Fibonacci and Lucas numbers.
12.1 SUMS OF LIKE-POWERS
Let r and s be the solutions of the quadratic equation . Then
so and . Consequently, and . Continuing like this, we can compute the various sums :
More generally, using PMI, Drain and ...