It is a very well known mathematical fact that the limiting ratio of consecutive terms of the Fibonacci sequence [F

_{0}=0, F

_{1}=1, F

_{n}=F

_{n-1}+F

_{n-2}] is Ļ=(1+ā5)/2 as nāā.

Suppose we generalize the definition of the sequence to:

F_{n}=**A**F_{n-1}+**B**F_{n-2}.

Find an expression for the limiting ratio of consecutive terms (in terms of A and B.)

Find formulas for A and B to make the limiting ratio any whole number N.

(In reply to

solution by Charlie)

Nicely done, Charlie!

A few things, starting with B = ((2N - A)^2 - A^2) / 4

a) Not clear why you did not just substitute A =1, to get

B = ((2N - 1)^2 - 1^2) / 4 = (4N^2 - 4N)/4 = N(N-1)

b) I do not consider A = N, B = 0 to be cheating. :-)

c) Not all values of A and N work in the above formula. Negative A and N do not give the expected result, I think. And at first I thought A = 0, B = N^2 would work, but this does not converge unless f(0) = f(1).