Evaluating Continued Fraction: \langle 1, 2, 1, 2, \ldots \rangle

[math_grl]

Ok I need to know which is the right answer for evaluating the continued fraction $$\langle 1, 2, 1, 2, \ldots \rangle$$?

Here's my work:
$$x = 1 + \frac{1}{2+x} \Rightarrow x^2 + x - 3 = 0$$ and by quadratic formula, we get $$x = \frac{-1 \pm \sqrt{13}}{2}$$ but we only want the positive root so I get $$x = \frac{-1 + \sqrt{13}}{2}$$ for my answer but the answer given was $$x = \frac{1 + \sqrt{3}}{2}$$, so I'm confused at which it is...

Moreover, I can't seem to find any other example except for $$\langle 1, 1, 1, \ldots \rangle$$ to see if I'm doing my computation right. Please help.

 

[Petek]

Your equation for x is incorrect. It should be

$$1 + \frac{1}{2 + \frac{1}{x}}$$

 

[math_grl]

that's embarassing.

 

[Petek]

You'll do better next time!

 

[blue_raver22]

I would advise you to check your work and make sure you have correctly applied the continued fraction formula. It is possible that you made a mistake in your calculations, which is why you are getting a different answer than the one given.

In terms of evaluating continued fractions, there is no one right answer as it depends on the specific continued fraction and the method used to evaluate it. It is possible that the answer given to you is using a different method or approach to evaluate the continued fraction.

I would also recommend looking for other examples of continued fractions to see if your computation method is correct. You can also try using a different approach to evaluating the continued fraction, such as using different formulas or algorithms.

Overall, it is important to carefully check your work and consider different approaches when evaluating continued fractions. There is no one definitive answer, and it is possible for different methods to yield different results.