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# Powers of Phi

### Phi has a unique additive relationship

The powers of phi have unusual properties in that they are related not only exponentially, but are additive as well.  We know that:

Ø2 = Ø + 1

Which is the same as:

Ø2 = Ø1 + Ø0

And this leads to the fact that for any n:

Øn+2 = Øn+1 + Øn

Thus each two successive powers of phi add to the next one!

 n Øn 0 1.000000 1 1.618034 2 2.618034 3 4.236068 4 6.854102 5 11.090170 6 17.944272

### Powers of Phi and its reciprocal

Another little curiosity involves taking phi to a power and then adding or subtracting its reciprocal:

For any even integer n:

Øn  +  1/Øn = a whole number

For any odd integer n:

Øn  -  1/Øn = a whole number

Examples are shown in the tables below:

for n = even integers

 n Øn 1/Øn Øn + 1/Øn 0 1.000000000 1.000000000 2 2 2.618033989 0.381966011 3 4 6.854101966 0.145898034 7 6 17.944271910 0.055728090 18 8 46.978713764 0.021286236 47 10 122.991869381 0.008130619 123

for n = odd integers

 n Øn 1/ Øn Øn - 1/Øn 1 1.618033989 0.618033989 1 3 4.236067977 0.236067977 4 5 11.090169944 0.090169944 11 7 29.034441854 0.034441854 29 9 76.013155617 0.013155617 76 11 199.005024999 0.005024999 199

The whole numbers generated by this have a relationship among themselves, creating an additive series, similar in structure to the Fibonacci series, and which also converges on phi:

 Exponent n 0 1 2 3 4 5 6 7 8 9 10 11 Result 2 1 3 4 7 11 18 29 47 76 123 199

Phi - The Golden Number
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