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Nested Square Roots


Date: 07/17/98 at 11:15:32
From: Natasha K.
Subject: Number series question

Dear Dr. Math,

I am in the 8th grade and I am on my school's math team. I have been 
practicing over the summer for next year's math league.

Here is a question that I came across that I couldn't solve. I didn't 
even know where to begin:

   Solve for n: 
   n = square root (6 + square root (6 + square root 6 + .. 
 
How can you prove what this series converges to?

Thank you.


Date: 07/17/98 at 13:07:20
From: Doctor Nick
Subject: Re: number series question

Hi Natasha,

The trick here is to notice that some of the stuff on the righthand 
side looks like the whole expression. That is:

   n = square root ( 6 + n )

One might object to this since it appears that the stuff inside the 
second square root has one fewer 6 than the whole expression; however, 
since there is an infinite number of 6's, the number of 6's in each 
expression is the same.

So, if we square both sides we get:

   n^2 = 6 + n

It might be clearer to get to this another way. Notice that if you 
square the right side and subtract 6, you get the expression you 
started with, n.

This gives us:

   n^2 - n - 6 = 0

so:

   (n-3)(n+2) = 0

Thus n is either -2 or 3, but n can't be -2, since n has to be positive 
(it's at least as big as the square root of 6). Thus n is 3.

You have to be careful with problems like this to make sure that the 
expression you're dealing with actually defines a number. When working 
with infinite expressions, it's especially important to be careful to 
know what's going on. It's not too hard to show that your expression 
does define a real number, but it takes a little care.

Notice that we could replace 6 by any other positive number. Can you 
work out a general formula for:

   square root(a + square root(a + squareroot(a + ...   

where a is any positive number? (You'll probably need the
quadratic formula).

Have fun,

- Doctor Nick, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory
High School Sequences, Series

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