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:
Fn=AFn-1+BFn-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).
re: solution
