Have you tried solving Challenge #10 yet? Go try it first if you haven’t. It’s not too hard, I promise!
All of these can be shown by just plugging in the values of and , but I’d like to show you some easier ways.
The first problem is easy — we know that is a solution to the equation ; that is, . Some simple algebra shows that . This is kind of interesting when you think about it. In English, this says that squaring is exactly the same as adding 1.
Problems 2 and 3 can be solved simultaneously. Since and are the roots of the polynomial , we know it can be factored as
Multiplying out the right side, we get
For this equation to be true, we obviously must have and .
There is not a nice slick way to prove #4 (at least, not that I am aware of). But doing it by plugging in values isn’t so bad:
Next up: we’re going to use these properties to prove some astonishing facts about the relationship between the golden ratio and Fibonacci numbers! Stay tuned!