Ratios of Consecutive Fibonacci Numbers
The Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21,.............
Here is the sequence formed by ratios of consecutive Fibonacci Number:
1 2 3 5 8 13 21
, , , , , , ,.....
1 1 2 3 5 8 13
Use a calculator to determine a decimal value for each fraction. Calculate the values for additional terms
and I think you will begin to see a pattern emerging.
We will use xn for the value of the ratio at position n.
x1  1
x2  2
In general, xn 
tn 1
and tn1  tn  tn1 i.e. any term is the sum of the two previous terms.
tn
 xn 
tn  tn 1

tn
x3  1.5 Calculate several more and it will become clear.
xn  1 
tn 1

tn
xn  1 
1

tn
xn  1 
1
xn 1
tn 1
In fact the ratios actually approach a fixed value... i.e. the ratios converge to a limit. Call that value, v.
As n gets very large, xn and xn 1 are virtually the same.
So the equation xn  1 
1
1
becomes v  1  .
v
xn 1
This becomes v 2  v  1 
v2  v  1  0. This produces the Golden Ratio... v 
1 5
2
1.618
It is quite interesting that the Fibonacci Sequence and the Golden Ratio are related.
There are some interesting applications to the growth patterns in plants....a spiral can be created using
squares whose side lengths correspond to the terms of the Fibonacci Sequence. The Greeks believed
that the most pleasing proportions for artwork, sculpture and architecture involved the Golden Ratio.