Binet Revisited

At this point, we return to Binet's Formula for calculating the nth Fibonacci number, armed with our new knowledge of the Golden Ratio. Let's take a look at the equation again:

At least one of the values in the smallest sets of parentheses should look familiar. Recall that back when we found the value of , we had to solve the quadratic equation

When we solved this equation, we got the following possible values for y:

These are the same as the inner values of Binet's Formula, and the first number is actually , the Golden Ratio.

So why can we calculate Fibonacci numbers by using the Golden Ratio? See the Proof page for this section for an explanation if why this works.