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Faldo: I think you misinterpreted FF's sequence; it is the square root of 1 plus the square root of (1 plus the square root of(1 plus etc.))). The square root of 1 plus the square root of one becomes the square root of 2, or 1.414+, and the increments become smaller and smaller as you add more extensions. So it has to converge somewhere rather than diverging to infinity, which it would do if it were the square root of 1 + (the square root of 1) + (the square root of 1). . . which is just another way of saying 1 plus 1 plus 1 all the way to infinity. See http://tinyurl.com/v0b2 which is pretty much the same thing as what FF set forth. I'm not as versed in this stuff as I may have been once, but it looks correct to me. I do wonder though why FF has discarded the negative solution to the quadratic equation, calling it an extraneous root. I don't know what that means and I would be loath to totally disregard it. TEd
TEd
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ok, now enough--we have a policy here about unacceptable words in posts and quadratic equation is likely to be one of those words..
this is a language board.. and we try and keep things warm and friendly--
now i will be the first to admit, i brought up the subject of math(s), but there really is no cause for that kind of language here!
--what we do in our spare time, in the privacy of our homes is one thing.. but really, is this the kind of language and subject we want to discuss here?
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Helen-- 
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Yup, I had me a cup of coffee and now realize that my purported rebuttal is all wet.
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Goodness, you startled me. I went back to my post secant for a sine that I had gone off on a tangent. But I will not be stopped! I still have a trig or two up my sleeve.
TEd
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hahaha, the series of square roots doesn't equal infinity. It resolves to the golden ratio, as I showed in the first part of the message. The problem is ya gotta wade through the junk.
It's not an easy thing to understand, really, as the solution relies on understanding how to manipulate recursive equations.
k
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Sometimes, either in the setup of a problem or during some manipulation you create these "extraneous" roots. I first encountered them back in algebra (1 or 2, I can't recall). I admit I was very skeptical at first, but gradually it's just something that I came to intuit without complete understanding. I'd like to explain the term clearly, but an adequate definition eludes me. Sometimes you look at something and you just know the answer. In this particular case we're square rooting a bunch of positive numbers - we know the result has to be positive. We probably 'created' the extraneous root when we squared both sides of the equation.
k
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all wet...nah, this stuff is really tricky. I remember getting into a big argument with an older chemist friend of mine about it - and he's a really smart guy. He went to a physics prof who backed me up. (Also, it's the same solution the author of the puzzle gave.)
k
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This kind of thing commonly happens when you're exponentiating by an even power, btw. Say y=sqrt(5) identically, therefore y^2 = 5, but if you solve for y, you get y = +/- sqrt(5). Our squaring operation just injected a false root into the solution.
You do have to be careful when you determine that a root is extraneous.
For example, computing a negative length for the side of a triangle might be an extraneous root, or it might mean that the problem has no solution, or it might be an indication that we've set up the problem wrong - or it could be an indication that the problem space is a little more complicated than we had supposed.
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we're square rooting a bunch of positive numbers - we know the result has to be positive. We probably 'created' the extraneous root when we squared both sides of the equation.Huh? I can't let this one by as it contradicts any and all maths I ever studied. Minus times minus gives plus - the root was there all the time and is not extraneous. BTW check out Ted's link page and you'll see that the value of the second solution is what my original Google ( http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html) mentioned as the 'closely related value which we write as phi with a small "p" ', and is the reciprocal of Phi as well as (Phi-1). Which means (I think, my head is now hurting as it's a long time since I did any of this!  ) that small-p phi is actually really the same ratio anyway. (Value n of the series):(value n+1) rather than (value n):(value n-1). So there's not even any reason to eliminate this solution from the quadratic equation. ..I was trying to write this post some 16 hours ago and at that point it killed not only my brain but my computer too... I was also planning to address FF's later concern about a negative value for the side of a triangle. I don't have a problem with this, or see that it is complicated. Assume that the side in question lies along the x-axis of a graph and has one end at the point of crossing the y-axis (0,0). The difference between positive and negative value for the length of the side is whether the triangle then lies to the right or the left of the y-axis. Put another way, which end of that side you have arbitrarily decided to define as (0,0). I see positive and negative as directional rather than absolute-value (consider also time and latitude as examples where we apply not just distance but also direction from an arbitrary point), but I think I'd better stop there as 1) I am out of my depth and 2) it is hard to type with a kitten on the keyboard!!! Bel, I apologise for hijacking your thread in the first place!!
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