I have attached the question I need help, question 5, and really not sure how to go about,
any help / tips appreciated
thank you
Begin with writing out a little bit of the continued fraction:
$\displaystyle y = s + \cfrac{1}{s+ \cfrac{1}{s + \cfrac{1}{s+\cfrac{1}{\ddots}}}}$
Now, you are asked to show that $\displaystyle y = s+\dfrac{1}{y}$. This is obvious. Just replace everything under the top 1 of the continued fraction by y (since it is equal to y).
So, how can you solve the rest of it? Well, solve for $\displaystyle y$. If you multiply everything by $\displaystyle y$, you get $\displaystyle y^2 = sy+1$. Treat $\displaystyle s$ as a constant and you have a quadratic of one variable. Solve for $\displaystyle y$. Then $\displaystyle x = 1 + \dfrac{1}{y}$. Just plug in whatever you get as your answer.
Huh?
I mean
$\displaystyle y = s + \cfrac{1}{s+ \cfrac{1}{s + \cfrac{1}{s+\cfrac{1}{\ddots}}}} = s + \cfrac{1}{\left(s+ \cfrac{1}{s + \cfrac{1}{s+\cfrac{1}{\ddots}}}\right)}$
According to the first equality, the part that is in parentheses in the rightmost equation is equal to $\displaystyle y$.