Wednesday, February 28, 2007
Back to Your Roots: Investigating Nested Radicals
Posted by Dave Marain at 12:15 PM
Labels: limits, nested radicals, quadratic, recursive functions
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8 comments:
(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.
TC
Thanks for the nice words... I would like something more interesting! That's why this is a 2-day investigation!!
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.
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.
You should introduce them to continued fractions next. You might also print out the MathWorld page on nested radicals.
eric--
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.
In (g), you may also ask what are all the possible values of such nested radicals.
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:
k----N
3----6
4---12
5---20
etc.
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!
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