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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 c016-math-001 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 c016-math-002 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 c016-math-003 approaches a limit, namely, 1.618033….

Table 16.1 Ratios of Consecutive Fibonacci and Lucas Numbers

n c016-math-004
1 1.0000000000 3.0000000000
2 2.0000000000 1.3333333333
3 1.5000000000 1.7500000000 ...

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