Proof by induction that *p* = 1/*p*-1 in golden rectangle exercise

Question

The initial rectangle's dimensions is L0 for the length and l0 for the width.

A golden rectangle can be obtained when it has the same proportions as the initial rectangle, so p = L0/l0

I am first asked to make relations between Ln+1 and ln, and then between ln+1 and, ln and Ln.

I got Ln+1 = ln and ln+1 = Ln - ln

Then I have to prove by induction that p= 1/p-1 before deducting p = 1+sqrt(5)/2

I'm totally stuck for how to prove this, what shall I do?

Thanks for your answers

Answer 1

Let $p_n = \dfrac{L_n}{l_n}$. Since $L_{n+1} = l_n$ and $l_{n+1} = L_n - l_n$ we have $$ p_{n+1} =\dfrac{L_{n+1}}{l_{n+1}} =\dfrac{l_n}{L_n - l_n} =\dfrac{l_n}{l_n p_n - l_n} =\dfrac{l_n}{l_n(p_n - 1)} =\dfrac{1}{p_{n}-1} $$ If $p_n$ is always equal to $p$ we get $$ p = \dfrac{1}{p-1} $$ as required.