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Square Root Trick

Date: 08/13/97 at 13:09:50
From: Daniel Lee
Subject: Square root

Is there a formula to calculate sqrt(x+sqrt(x+sqrt(x+... ? 

Date: 08/18/97 at 14:40:13
From: Doctor Rob
Subject: Re: Square root

Yes, there is a very cute trick to figure this out.

If x = 0, presumably this value is 0.  If x < 0, I'm not sure that
this makes sense, but perhaps even so.  We assume x > 0.  

Let a = sqrt(x+sqrt(x+sqrt(x+...))).  Then square and subtract x.  
You get

    a^2 - x = sqrt(x+sqrt(x+...))
            = a

This is a quadratic equation for a. When you solve it for a, you
get two solutions. You must reject one of them as extraneous. That
leaves you with the formula you seek.

-Doctor Rob,  The Math Forum
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Associated Topics:
High School Exponents

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