Find $r$ in a geometric progression

Question

In a G.P ($b_n > 0) $ $$b_n = b_{n-1} + b_{n-2} $$ $n \ge 3 $

find $r$ of G.P.

Answer 1

$$b_n = \frac{b_n}{r} + \frac{b_n}{r^2} \Rightarrow b_n = \frac{b_nr + b_n}{r^2} \Rightarrow \frac{b_n(r+1)}{r^2} \Rightarrow 1 = \frac{r+1}{r^2}.$$

$$r^2 -r -1 = 0 \Rightarrow r = \frac{\sqrt5+1}{2}.$$

In the end, you have to take the quadratic formula. In the beginning $$b_{n-1} = \frac{b_n}{r}.$$

I hope you understand. If there is any question, be free to ask.