The famous golden ratio is

g=

Ö5 + 1 2

. Patrick and Andrei showed that

g2 = (Ö5 + 1 )2 4 = (6 + 2Ö5) 4 = (Ö5 + 3) 2 = g + 1

.

Marcos considered the equation x² = x + 1. By completing the square,

(x - 1/2)2 = 5/4

he obtained the solutions

x= 1 ±Ö5 2 .

So g is one solution of this equation and hence g²=g+1.

Both Andrei and Marcos then experimented a bit to get familiar with the problem...

Multiplying by g

g2 = g+1 g3 = g2 + g = 2g + 1 g4 = 2g2 + g = 3g + 2 g5 = 3g2 + 2g = 5g + 3 g6 = 5g2 + 3g = 8g + 5 ¼ .

Marcos then made the conjecture that a_n+1 = a_n + a_n-1 and these coefficients are the Fibonacci sequence. Similarly for the b_n and went on to prove this conjecture using the method of mathematical induction.

Patrick used the result g²=g+1 to find a pattern in the coefficients for gⁿ = a_ng + b_n as follows. Multiplying by g gives

gn+1 = ang2 + bng = an(g+1) + bng = (an+bn)g + an.

Thus a_n+1=a_n+b_n and b_n+1=a_n. Hence a_n+1=a_n + a_n-1. This is the Fibonacci sequence. Since the first terms are a₁=1 and a₂=1 so gⁿ=a_ng+a_n-1 where a_k is the k-th term of the Fibonacci sequence.

Then Patrick went on to prove by induction that

an=

an-bn Ö5

where a and b are the solutions of the quadratic equation x² = x + 1. Now for n=1

a- b = 1+Ö5 2 - 1-Ö5 2 =Ö5

and dividing this by Ö5 gives the value of a₁=1 showing that the result is true for n=1. For n=2

a2 - b2 Ö5 = (a + 1)- (b+1) Ö5 = 1

as above giving the value of a₂=1 showing that the result is true for n=2. Assume the result for n=k and n=k-1 and using the fact that a² = a+1 and b²=b+1 then

ak+1 = ak + ak-1 = ak-1(a+1)-bk-1(b+1) Ö5 = ak+1-bk+1 Ö5

Hence by the axiom of induction the result is proved. Thus

gn = an-bn Ö5 g + an-1 - bn-1 Ö5 .