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of troy, since you can remember it's called the Fibonacci Sequence (which I only remember when you point it out - passive, not active) don't give me that about your lack of maths!!  I wasn't aware of a Greek letter for the 'final value' this ratio tends to, but a quick google shows it is phi, and also that: The golden ratio [approximately] 1·618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a greek letter Phi . The closely related value which we write as phi with a small "p" is just the decimal part of Phi, namely 0·618034 ...why, as a native NY'er, are you talking about Alevel maths, which is a British qualification?
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why A levels and further maths?-- i have been reading the curious incident of the dog in the night--and have thinking how funny maths sounds.
i understand the term, (and its a rather general term) and i couldn't think of any american english term that was general (for describing both HS or college math) and i expected most who read here to understand it.
i suppose i could have said, trig, calculus, and AP math, or some other sets of terms, but A levels work..
Oh, 1.61.. not 1.63.. well not bad, not bad.. and i totally forgot the reciprical until you posted.. (well it was late at night)
i do read math for fun.. but i am not good enough (at most math) to learn it well without applying disipline and work, and well i read it for FUN. so i have some understanding (unlike pi--which i know, and understand in my heart of hearts, because i was made to do the work, and in working through, i really learned it.)
(i know moebius strips and klien bottles well.. i am much better at mental manouvering 3 d object, --and as fun task, i knit my self a moebius strip scarf. it has one edge (the cast off edge, one side, and was knit in one piece (not knit as a flat strip of knitting, then sewn together)
it was fun imagining a knit moebius, mentally unraveling it, and figuting out how to start with out a cast on edge.
(and then i did it.) PM if you want links to the photos.
i know a bit about golden numbers and golden ratios, because they particularly facinate my son, and he used to talk about them all the time, (and i read a bit so as to be able to understand him, and his facination.)
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Ooh! Ooh! [waving hand wildly e] the curious incident of the dog in the night Good book! And, phi: "The DaVinci Code"! And, i read a bit so as to be able to understand him, and his facination We (human beings) do do a lot for our kids, don't we? That is why I learned (and watched) all the Sesame Street characters when my kids were little, and why I listened to rock radio stations when they became teenagers. I'm glad you were able to share your son's interest in math; I have not been able to make myself get "into" every interest of my kids': namely my daughter's inexplicable interest in politics [Y-A-W-N] and my son's knowledge of various calibers (all I need to know is that it's a gun and it shoots; I figure whoever gets me set up will put the right size in.) it was fun imagining a knit moebius, mentally unraveling it, and figuting out how to start with out a cast on edge. Now, I think that is just the coolest thing! [applause] But...when you go to put it on, how do you know where to start? 
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> my daughter's inexplicable interest in politics
That deficit, like others, had to end up somewhere ;)
heh, does B like Shakespeare too?!
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I agree, Dog in the Nighttime is a great book. I don't agree that maths sounds funny - I think math sounds funny!  and as for knitting, I can't do that either. of troy, you are (unintentionally I assume!) exposing all my weaknesses... 
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does B like Shakespeare too?! YES, darn you! And he's good at performing it, too.
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well knitting is (i think, and i am not alone) very mathmatical.. (its all done in binary, and a bit of understanding boolian doesn't hurt --of course you can learn to knit with out consciously learning math(s) but knitting is so much more fun if you know both--
knitting a the new rage (i've been knitting 40+years, and finally, its in style(for now). when it becomes passe again, i will still be knitting.
--in the UK the rage is radical knitting (the newest rage is the knit purse that looks like a hand grenade) and a show at the London crafts museum.. ( could look up a link to the review in the gardian.. )
but a lot of radical knitting is not really new
what is radical anyway? anything your nana didn't knit? suppose your nana was radical in her day and age? --then what?
willie warmers --yes they are what you think they are--are 'radical' but the patterns for making them date from the 1970's or so. (or perhaps early--i remember them from then, and they might not have been a new idea then!)
some hippies were into 'return to the earth, new age, simpler living crap--and they knit them.. so they are hardly radical.
sometimes, everything old is new again.
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> everything old is new again
S'what I say every day when I drag my bones out of bed, Helen :)
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At http://geocities.com/elbillaf/ratio.txt I have a non-rigorous proof that the ratio of successive elements tends towards phi. (It's towards the bottom half of the page and is mentioned in the context of puzzle.) k
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I didn't wade through that whole "proof" but your original series of linked forever and ever square roots equals infinity so I don't know what you expect to "prove" by saying that infinity - 1 = infinity.
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