1) Under what conditions will F1 and F2 be equal?
2) If F1 and F2 are equal, what is their value?
I'll rename, for simplicity: the first continued fraction I'll call "x", and the other I'll call "y".
Note that x = A + B/(A + (B/(A + B/...)))) = A + B/[A + B/(A + (B/(A + (B/...)))) = A + B/x. Similarly, y = B + A/y.
For these to be equal, we must have A + B/x = B + A/y. What can you get from this?
Hello, dxdy!
I have a good start on it . . . I'll let you finish it.
Let and be distinct positive real numbers and consider the continued fractions:
. .
1) Under what conditions will ?
From [1]: .
. . Quadratic Formula: .
From [2]: .
. . Quadratic Formula: .
. . . . . .
. . . . . .
Square both sides:
. .
. . . .
Square both sides:
. .
. .
This simplifies to: .
. . Factor: .
. . Factor: .
Since , we can divide by
Can you finish it?