Chapter 2.2, Problem 38MS

Beyond Fibonacci. Suppose we create a new sequence of natural numbers starting with 0 and 1. Only this time, instead of adding the two previous terms to get the next one, let’s generate the next term by adding 2 times the previous term to the term before it. In other words: $G_{n+1}=2G_n+G_{n-1}.$

Such a sequence is called a generalized Fibonacci sequence. Write out the first 15 terms in this generalized Fibonacci sequence. Adapt the methods that were used in this section to figure out that the quotient of consecutive Fibonacci numbers approaches $(1+\sqrt{5})/2$ to discover the exact number that $G_{n+1}/G_n$ approaches as n gets large.