More continued fractions! (yay)

fbellman

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Jan 8, 2006
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Thanks a lot for everyone's help in my LAST post on continued fractions... hope this is the last post I'll have to make on this topic!

So, here's a general continued fraction:

continuedfraction3hj.jpg


I'm supposed to consider values of k that are not 1 and 2, and determine a generalized statement for the exact value of any such continued fraction. I figured out that this exact value is:

value5oo.jpg


First off, is the above correct? I determined this by using the quadratic formula and the equation (t sub(n))^2 - kt sub(n) - 1 = 0 (I did this because of the help in my last post, thanks!).

Secondly, I need to figure out this: For which values of k does this generalized statement hold true? How would I go about figuring this out? Plugging in different numbers and making up a table?? That's my only idea. :(

Thanks a lot for your help!
 
Yup, it is correct. It is valid for all real k because k^2+4 is positive and the sqrt exists.
 
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