Something that I see a lot at math competitions or on Brilliant is a problem asking to find the value of for some value. I'm going to talk about how to solve that "nested radical" as well as "nested" fractions.
First I'm going to talk about how to solve a problem such as finding the value of Let You may notice that you are adding to and taking the square root of this. I'll demonstrate this with some parentheses.
For simplicity, we're just going to use You may notice that this is the same thing as saying that Let's try solving that.
We can eliminate the solution because the nested radical obviously is non-negative. Thus, we can say that
Now we can use a similar line of logic to find a general form for Let that equal We will make use of the same procedure, but using the quadratic formula to solve for
Since we are dealing with real numbers, the value under the radical cannot be negative (except for .) Thus, we can say that
Now let's use the same logic to derive a formula for nested fractions of the form as well as continued fractions where the numerator is not necessarily
First, the obligatory demonstration of such a fraction.
Again, we can throw out the negative solution and say that
Moving on, we can derive a general formula for finding allowing this expression to be equal to
So a summary of what we have proved here:
If you want some practice problems, here's two.
Find a closed form for
This calculus problem
Find all values of for which is an integer.
I hope this helps you out! Please feel free to share what you think in the comments. Thank you very much!