Inspired by comments on Jil's most recent post, let's have a little quiz from the teacher. Seeing that most of you have had some basic math classes, this should be cake. Thanks to R, we all now know the fibonacci sequence begins with 1,1,2,3,5,8,.... Each number in the sequence is simply the sum of the previous two (after being given the first tw0). That is, we begin with f(1) = 1 and f(2) = 2, and then f(3) = f(1) + f(2) = 1+1 = 2. More generally, f(n) = f(n-1) + f(n-2).
Now here is the question:
Form a new sequence X of numbers by dividing each number in the fibonacci sequence by the previous number in the sequence, starting with the second number in the fibonacci sequence.
e.g. X(1) = f(2)/f(1), X(2) = f(3)/f(2), ..... X(n) = f(n+1)/f(n).
This new sequence X converges to what number as n approaches infinity?
The rules of this quiz are that you can only use a paper, writing utensil, and calculator...no spreadsheets or web searches!!! The first person to post the correct answer wins something.
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3 comments:
I figured they only taught you how to build space shuttles and such at MIT, not minor math calculations like converging sequences.
As n approaches infinity (i'll call it oo for the sideways "8") f(oo)=f(oo-1)+f(oo-2)
But since you're already at such a high number, f(oo-1) is one less oo than f(oo). So f(oo+1)/f(oo)... the numerator will always have one more 'oo'. But as the number gets bigger and bigger, the number will converge to 1... but then I start plugging numbers in and it seems to approach 1.6 instead. Hmmmm.... I must have taken a wrong leap... going back... As n becomes infinite, wouldn't f(oo)=2oo? So f(n+1)=3oo. So X(oo)=3oo/2oo = 3/2 = 1.5!
Is that it?
You are correct in your initial assumption that oo = 2oo = 3oo and so forth. However, the 1.6 number is what you are going for. After I posted it, I tried to figure it out and it actually took me a while. I'll give it another day or two and post the proof. A hint is to try rearranging/substituting/etc. the f(n+1)/f(n).
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