Value of an indefinitely long expression : Quadratic Equation Approach

Question

I came across this problem and have been able to get the correct answer for this although I tried many times on my own.

$1st$ Approach : $8+\frac{8}{8+\frac{1}{8+y}} = y$ and when I solve this I get to the quadratic equation i.e. $584 + 7y = 8y^2$ which does not lead to the correct solution as the roots for this quadratic equation are $y = \frac{7}{16} \pm \frac{\sqrt(18737)}{16}$ .

$2nd$ Approach : $8+\frac{8}{8+\frac{1}{y}} = y$ and when I solve this I get to the quadratic equation i.e. $8+ 71y = 8y^2$ which also does not lead to the correct solution as the roots for this quadratic equation are $y = \frac{71}{16} \pm \frac{\sqrt(5297)}{16}$ .

Please help me on this ! I am stuck with this problem for a while.

Thanks in advance !

PS : This question is a MCQ from where I picked up this question and the options avaliable are

1. $6\frac{2}{3}$
2. $4+\sqrt17$
3. $8+\sqrt8$
4. 9

I am expecting an analytical and step-by-step solution as I was trying to do.

Answer 1

I am assuming that there is a typo in the question as the pattern is not obvious.

Let $$8 + \frac{\color{red}{1}}{8+\frac{1}{8+\ldots}}=y$$

$$8 + \frac1y = y$$ $$y^2-8y-1=0$$

$$y = \frac{8 \pm \sqrt{64+4}}{2}$$

Since $y > 0$, $y = \frac{8 + \sqrt{68}}{2}=4 + \sqrt{17}$

---

I did consider other possibilities but I did not hit one of the options: $$8 \left( 1+ \frac1{8 + \frac1{8 + \dots}}\right) = y$$

Now, first, let's study what do we get from $\frac1{8 + \frac1{8 + \dots}}=z$

$$\frac1{8+z}=z$$

$$z^2+8z-1=0$$

$$z=\frac{-8 +\sqrt{64+4}}{2}=-4 + \sqrt{17}$$

$$8 (-3+\sqrt{17})=y$$