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Fibonacci Sequence and PHI

The Fibonacci sequence is a famous sequence of numbers starting with 1,1 where each term (from the third onwards) is the sum of the previous two terms. The first few numbers in the sequence are as follows:

1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,
46368,75025,121393,196418,317811,514229,832040,1346269….

which quickly become very large.

The Fibonacci numbers, fr can be defined by
f1 = f2 = 1
and
fk = fk-1 + fk-2 for k≥3

PHI the golden ratio

PHI or Φ is said to be the golden ratio since so many things in nature seem to naturally arrange themselves in this ratio. It is approximately equal to 1.618033988749895.

Phi is also the positive solution to the equation

x2 = 1 + x

which has the irrational solution

phi = (1+√5)/2

Interestingly an irrational number is one which cant be written as the ratio of two integers so the golden ratio is not in fact a ratio meaning ratios in nature can only become very close to it but cant actually equal it.

PHI and Division of Fibonacci numbers

It is interesting to calculate the ratio of consecutive numbers in the Fibonacci sequence. These ratios quickly approach the golden ratio know as PHI or Φ.

The start of list of value obtained from these divisions are:
1
2
1.5
1.6666666666666667
1.6
1.625
1.6153846153846154
1.619047619047619
1.6176470588235294
1.6181818181818182
1.6179775280898876
1.6180555555555556
1.6180257510729614
1.6180371352785146

Calculation of PHI from limit of Fibonacci divisions

If we suppose that the ratio of consecutive terms in the Fibonacci sequence do approach a limit we can use this to find the value of phi.

IF we denote the nth Fibonacci number by fn and the nth ratio as rnthen

rn = fn+1 / fn

But using the definition of a Fibonacci number:

fn+1 = fn + fn-1

then the ratio is

rn = (fn + fn-1 )
fn

Which can be simplified to be

rn = 1 + fn-1/fn

but fn-1/ f = 1/rn-1

so

rn = 1 + 1/rn-1

Now supposing that the ratios tend to a limit p as n tends to infinity then p is the solution of the equation

p = 1 + 1/p

which can be re-arranged to give

p2 – p – 1 = 0

and can be solved using the quadratic equation to give

p = (1 + √5)/2
and
p = (1 – √5)/2

The first of these solutions happens to be the golden ratio PHI or Φ

If anyone has a proof that these ratios do infact approach a limit please include it in the comments of this post.

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