I solved both of them, slowly, in a sleep-deprived state. For the continued fraction, I first tried doing successive approximations to see what the answer "should" be... when I got 1.41 I figured that it was probably the square root of 2. So the next thing I did was to try squaring the expression, which wasn't exactly helpful, but it did lead me to notice that the continued fraction contained itself so I could use the algebra trick that Luke_A_Somers used.