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# Continued fraction.....

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Evaluate the following "continued fraction" : {1; 1, 2, 1, 2, 1, 2, 1, 2.........etc.}. Thanks for help.

Sep 21, 2017

#1
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Can you please rephrase that? I'm not sure what {1; 1, 2, 1, 2, 1, 2, 1, 2.........etc.}. Also, I think the mathy term you are trying to say is "repeating decimal", or "continued decimal".

Sep 21, 2017
#2
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No, it is called "continued fraction" of a well-known number: It means:

1 + 1/(1 + 1/(2 + 1/(1 + 1/2 +.......etc.)))

Sep 21, 2017
#3
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That is right! This is called "continued fraction" as the questioner rightly says so:

We can represent it as follows: x = 1; 1, 2, 1, 2, 1, 2.......etc., which,in turn, can be written as:

1+ 2 / {1+x} = x, solve for x

2/(x + 1) + 1 = x

Bring 2/(x + 1) + 1 together using the common denominator x + 1:
(x + 3)/(x + 1) = x

Multiply both sides by x + 1:
x + 3 = x (x + 1)

Expand out terms of the right hand side:
x + 3 = x^2 + x

Subtract x^2 + x from both sides:
3 - x^2 = 0

Subtract 3 from both sides:
-x^2 = -3

Multiply both sides by -1:
x^2 = 3

Take the square root of both sides:
x = sqrt(3) or x = -sqrt(3). Since the continued fraction has all positive signs, then the correct is: sqrt(3).

Sep 21, 2017
#4
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$$x=1+\dfrac{1}{2+\frac{1}{1+...}}\\ 1+\dfrac{1}{x}=1+\dfrac{1}{1+\frac{1}{2+...}}\\$$

First we let x = {1;2,1,2,1,...}, then 1+1/x = {1;1,2,1,2,1,2,...}

Then we find the value of x.

$$x=1+\dfrac{1}{2+\frac{1}{1+...}}\\ x=1+\dfrac{1}{2+x}\\ (x-1)(2+x)=1\\ x^2+x-3=0\\ (x+\dfrac{1}{2})^2=\dfrac{13}{4}\\ x+\dfrac{1}{2}=\pm\dfrac{\sqrt{13}}{2}\\ x=\dfrac{-1+\sqrt{13}}{2}\text{ or }x=\dfrac{-1-\sqrt{13}}{2}(rej.)\\$$

The value of the original expression is 1+1/x which is:

$$1+\dfrac{2}{\sqrt{13}-1}\\ =\dfrac{\sqrt{13}+1}{\sqrt{13}-1}\\ =\dfrac{14+2\sqrt{13}}{12}\\ =\dfrac{7+\sqrt{13}}{6}$$

:D

~The smartest cookie in the world.

Sep 21, 2017
edited by MaxWong  Sep 21, 2017
#5
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x=1+\dfrac{1}{2+\frac{1}{1+...}}\\ 1+\dfrac{1}{x}=1+\dfrac{1}{1+\frac{1}{2+...}}\\

$$x=1+\dfrac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}}\\ x-1=\dfrac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}}\\ x-1=\dfrac{1}{1+\frac{1}{2+(x-1)}}\\ x-1=\dfrac{1}{\frac{2+x-1+1}{2+x-1}}\\ x-1=\dfrac{1}{\frac{2+x}{1+x}}\\ x-1=\dfrac{1+x}{x+2}\\ (x-1)(x+2)=x+1\\ x^2+x-2=x+1\\ x^2=3\\ x=\pm\sqrt3$$

As our guest said, all the signs are positive so the answer is    $$\sqrt3$$

.
Sep 21, 2017