THE GOLDEN RATIO
He that holds fast the golden mean,
And lives contentedly between
The little and the great,
Feels not the wants that pinch the poor
Nor plagues that haunt the rich man's door,
Embittering all his state.
–William Cowper, English poet (1731–1800)
What can we say about the sequence of ratios of consecutive Fibonacci numbers? Does it converge? If it does, what is its limit? If the limit exists, does it have any geometric significance? We will pursue these interesting questions, along with their Lucas counterparts, in this chapter.
16.1 RATIOS OF CONSECUTIVE FIBONACCI NUMBERS
To begin with, let us compute the ratios of the first 20 Fibonacci and Lucas numbers, and then examine them for a possible pattern; see Table 16.1. As n gets larger and larger, it appears that approaches a limit, namely, 1.618033….
Table 16.1 Ratios of Consecutive Fibonacci and Lucas Numbers