In Question 4.2.21, we found that the ratio between successive terms in the Fibonacci sequence seemed to be approaching $1.618$. Maybe it’s approaching the golden ratio, $\phi = (1+\sqrt5)/2$. Assuming there were a constant ratio between successive terms, and each term is the sum of the previous two terms, find the ratio. Show answer

Let’s call the ratio $r$, then if a term is $t$, the next term is $rt$, and the one after that is $r\times rt$ (assuming a constant ratio), or $t + rt$ (assuming that each term is the sum of the previous two). Therefore, \begin{align*} r^2 t &= t + rt\\ r^2 t - rt - t &= 0\\ r^2 - r - 1 &= 0\quad\text{if }t\ne 0\\ r &= \frac{1\pm\sqrt{5}}{2} \end{align*} One of these solutions is negative, and we know that the Fibonacci numbers don’t go negative, so the only sensible solution is $(1+\sqrt5)/2$, the golden ratio.