Continuous Fraction/ 连分数

2010-01-15 4:22 am

I am stuck on a question about a continued fraction, which is to find the EXACT value of:

[5;6,5,6,5,6,5...]

can somebody do that for me please?

If you are not sure what that means, go to this website and take a look
http://zh.wikipedia.org/wiki/%E8%BF%9E%E5%88%86%E6%95%B0

To give an example, I have done another one that is

sqrt(6+sqrt(6+sqrt(6+sqrt(6...))))

and my solution to that one was

let x be the number
so x^2 = 6+sqrt(6+sqrt(6+sqrt(6...))))
and that means x^2 = x + 6
and I solved the quadratic equation of x^2+x+6=0

I have to hand this in tomorrow so please help me!!

回答 (1)

用戶9112

2010-01-15 5:08 am

✔ 最佳答案

x = 5 + 1/[6 + 1/(5 + ...)]}藍色是重複部份
x = 5 + 1/[6 + 1/x]
x = 5 + x/(6x + 1)
(x - 5) = x/(6x + 1)
(x - 5)(6x + 1) = x
6x^2 - 29x - 5 = x
6x^2 - 30x - 5 = 0
x = [30 +/- √(900 + 4*6*5)] / (2*6)
x = 5/2 + (√1020)/12捨去負數

2010-01-15 09:21:10 補充：
因為題目中的分數是一正數

👍 0 👎 0