Wednesday, February 28, 2007

Back to Your Roots: Investigating Nested Radicals


Totally_clueless said...

(g) is a really nice generalization. Do you expect an answer of the sort 4N+1 is a square, or something more interesting and quite cool, IMO.


Dave Marain said...

Thanks for the nice words... I would like something more interesting! That's why this is a 2-day investigation!!

Anonymous said...

This one look fun.

And now I'm off to fix my printer so I can print it out and take it with me this evening.

jonathan said...

I have run this problems a bunch of times, with adults and kids. I use improving numerical approximations, leading to a conjecture, leading to some algebra...

That's with 2's. I'll extend to 1's or to the generalization. 4N+1 is a square? Hm. I think about it in slightly simpler terms.

Eric Jablow said...

You should introduce them to continued fractions next. You might also print out the MathWorld page on nested radicals.

Dave Marain said...

you anticipated my next problem - now i have to change it! Seriously, I already referred to the MathWorld article in an earlier comment; it is excellent and provides some fascinating general formulas. The challenge is to develop a sequence of questions that guide students toward understanding a concept culminating in a more sophisticated question that extends their thinking. This is what i enjoy doing the most.

bd said...

In (g), you may also ask what are all the possible values of such nested radicals.

Dave Marain said...

ok, I waited a few days and no one has yet offered an expression for N that would lead to an integer value for the infinite sequence of nested radicals...
I'll start with the answer:
N = k(k-1) where k = 2,3,4,...

Here's a derivation:
Let x denote the value of the nested radical expression. Then x satisfies the equation
x = sqrt(N+x) which leads to
x^2 - x - N = 0, which leads to the solution
x = (1+sqrt(1+4N))/2. If this expression is to be an integer, call it k (rather than x), then
1+sqrt(1+4N) = 2k -->
1+4N = (2k-1)^2 -->
1+4N = 4k^2-4k+1 -->
N = k^2-k = k(k-1). QED!
For example, if k = 2, then N = 2.
Other values:
Remember, k denotes the actual value of the nested radical expression.
I know someone will generalize this further!
Note that N = 1 leads to
x = (1+sqrt(5))/2, the golden ratio!