15. Solve .
solutions are
i think.
I'm interested in a method more than solution!
it was basically a bit of guesswork at first
i factorised 1680, worked out what the product of the four terms must be if integers, and got the two solutions
. the rest just was expanding out, shifting things around, and using the quadratic formula. certainly not a very effective method of solving the problem, it must be said.
in fact, any continued fraction that repeats itself can be solved using the quadratic formula. can you prove this?
If the continued fraction repeats, at some point in the continued fraction tower you will place an x instead of continuing it forever. Now to simplify the tower with x, you will perform addition and multiplication, so it simplifies into an expression of the form (ax + b)/(cx + d) which consequently results in a quadratic equation.
that's pretty much it. it
is possible to put this into a formal proof, but it takes a lot of time and effort to write it up.