Φ is the positive solution of the equation
and it can therefore be approximated by the sequences generated by the algorithms of calculation of the zeros of the continuous functions applied to the function
In particular, since
for the theorem of Bolzano Φ belongs to the interval [ 1;2 ].
Using the method of the secants on this interval we have
The terms of this sequence are
We see again the numbers of Fibonacci.
If we use the method of the tangents (Newton-Raphson) on the same interval [ 1;2 ] we have
The terms of this sequence are
The terms of the fractions in the sequence are again numbers of Fibonacci.