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 [280].
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 ...
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