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### Nested Square Roots

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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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