If you actually divide successive terms in the Fibonacci sequence, the quotients settle down to a specific value as you can see in the table below.
1/1 
= 1 
2/1 
= 2 
3/2 
= 1.5 
5/3 
= 1.666... 
8/5 
= 1.6 
13/8 
= 1.625 
21/13 
= 1.61538... 
34/21 
= 1.61904 
55/34 
= 1.61764 
89/55 
= 1.61818 
144/89 
= 1.61798 
233/144 
= 1.61805 
377/233 
= 1.61802 
610/377 
= 1.61803 
The quotients approach a number called the Golden Ratio whose value can be calculated algebraically. Using conventional subscript notation, the Fibonacci sequence is defined:
f_{1} = f_{2} = 1, f_{n+1} = f_{n} + f_{n1}, n = 2, 3, ....
Now, the Golden Ratio is usually called Ø. We say f_{n+1}/f_{n }→ Ø as n → ∞. Noting that both f_{n+1}/f_{n} and f_{n} /f_{n1} have the same limit Ø as n → ∞, the calculation goes like this:






Now replace the two terms: f_{n+1}/f_{n} with the variable x and f_{n1}/f_{n} with 1/x.










This quadratic equation has the two solutions:




The first solution is the Golden Ratio Ø, and the second is just  1/Ø !
Why is this ratio golden? That's really hard to say. It is definitely related to the pleasing nature of the proportion and the uncanny way it pops up in architecture and art; for example, we find the ratio in:
Many people claim that the ratio also appears in the Parthenon, in Athens. Unfortunately, it is not possible to determine the original measurements of the building due to its partial destruction over the centuries.
Lesson Plan 4