Like the previous page, this page could easily fit in the Slick Math! section of the site, because it has a math problem which, at a glance, look fairly intimidating, but is actually quite simple and elegant to solve. 

Take a look at the following continuing radical expression.
 

If you read the previous page, you probably have a good idea how to tackle this problem. We start by setting the expression equal to R.

Notice, now that the part of the expression which is circled in red is identical to the entire expression:
 

From this we conclude that R² - R - 1 = 0. I sure hope this looks familiar to you, because it is exactly the same quadratic equation that we had on the previous page! And, of course, it still evaluates to The Golden Ratio!

1.

In the continued radical shown, what would the value be if you replaced all the ones with twos?

2.

If you stopped the continued radical by replacing the contents of the fifth square root sign with a one, use your calculator to find the value.

3.

Try the same thing, but stop the continued radical after the sixth square root.

4.

How far out do you need to calculate this to convince yourself that it really does equal the golden ratio?