ADDITIONAL FIBONACCI AND LUCAS FORMULAS
In Chapter 5, we found explicit formulas for both and , namely, Binet's formulas. In this chapter we will derive additional explicit formulas for both .
8.1 NEW EXPLICIT FORMULAS
To begin with, we will conjecture an explicit formula for . To this end, recall that , so, as n gets larger and larger, ; and hence . So we compute the value of for the first ten values of n and then look for a pattern:
The pattern might not be obvious; so we will go one step further. Add 1/2 to each, and see if a pattern emerges:
A pattern, surprisingly enough, does ...