Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I came across the following in a quiz contest qualification test: $$x = 2 + {1\over 2+ {\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}}$$ Find the value of: $$\frac{3x^2+5x -3}{2x^2 -4x+5}$$ Now, I know that solving the equations is not as important as working out the logic to solve them. So can someone please explain the logic to me?

share|cite|improve this question
up vote 7 down vote accepted

From the equation of $x$ you can get $x=2+\frac{1}{x}$ i.e. $x^2=2x+1$ Then after plugging into given rational function you get

$$\frac{3x^2+5x-3}{2x^2-4x+5}=\frac{6x+3+5x-3}{4x+2-4x+5}=\frac{11x}{7}.$$ Now solve quadratic equation for $x$ and plug it in here.

share|cite|improve this answer

From the continued fraction you get $$x = 2+\frac1x \qquad\text{ or }\qquad x^2= 2x+1$$

You can use this expression for $x^2$ to simplify the numerator and denominator of the fraction separately (I get $\frac{11x}{7}$ but might have made a mistake somewhere, so don't trust that). Then it's just a matter of solving the quadratic and inserting the solution for $x$.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.