4. Secants and tangents.

Φ is the positive solution of the equation

fig. 1

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

fig. 2

In particular, since

fig. 3

for the theorem of Bolzano Φ belongs to the interval [ 1;2 ].

Using the method of the secants on this interval we have

fig. 4

The terms of this sequence are

fig. 5

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

fig. 6

The terms of this sequence are

fig. 7

The terms of the fractions in the sequence are again numbers of Fibonacci.