one for all ...
What is 1/(1 + 1/(1 + 1/(1+ ..... ?
Labels: mathemagic, SharedPuzzle
posted by Chris at
9:40 AM
Tuesday, August 18, 2009
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15 Comments:
.. 1
that does not make sense, there is no equality sign there, so you are basically telling us to put in a random number.
This post has been removed by the author.
Rick, wrong, try again.
Anonymous, I can't see why you are confused. I'm simply asking you to state the value of the infinitely deep continued fraction, not to write an equation. If you wish to write it as an equation, that's fine, I can cope with that.
Where are you proposing to put the random number into? That statement beats me for sure.
(1+sqrt(5))/2 very non intuitive
U R lukewarm.
how u mean lukewarm??
thats pretty exact, just not the complete solution set...
OK both of your answers are wrong.
But they LOOK similar to the correct answers.
(squareroot(5)-1)/2
Anonymous - you got it.
The solution: Assuming that a solution exists, call the associated value x. Then x = 1/(1+x) => x^2 + x - 1 = 0
=> x = (-1 +/-Sqrt(5))/2
If now check by plugging approximations to these solutions into the the recursion, x[n+1] = 1/(1+x[n]), only the solution with the +root of 5 returns the original value.
So I deem that (despite my initial belief) that there is in fact only one acceptable solution.
Not written clearly, but I'm sure you'll get my drift.
A slight variation of the original problem: 1+1/(1+1/(1+... would have returned the Golden Mean.
That was a roundabout way of saying the continued fraction only has one limit point.
okay, i admit i made a sign error while i made the eq.
But nice prob chris
Thanks tha b.H. Another golden oldie messed up (by me). I knew you'd crack it.
I wish I'd examined it better before posting, I only really started thinking about it when writing up the solution. I would have been clearer about the limit concept up front. To wit, I didn't know one solution was dodgy when I posted.
assuming that the question goes on forever I would think the answer would be 1/2.
1+1=2
and forever on the answers inside the parenthensis would be 2
2/2 = 1 for all
The beggining 1/2=.5
So the answer is .5
comment posted by:
Vincent Jumper
Vincent, You haven't correctly interpreted the original expression. You do "(" before "/" before "+".
Nowhere in the expression can you do 1 + 1 = 2. In fact you cannot simplify any sub-expression in the infinitely deep continued fraction.
You have to use the trick of noticing the expression contains itself. Let the value of the expression = x. Then you can write x = 1/(1+x). See solution above for the rest.
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