Hi Camilo,
programmig the Fibonacci Serie is a very standard exercise - and that is probably the origin of your question. There are basically two ways to do it:
One can use the recursion directly:
Clear[recFib];
recFib[n_] := recFib[n] = recFib[n - 1] + recFib[n - 2]
recFib[0] = 0; recFib[1] = 1;
This is elegant but in no way effective! (The double definition "recFib[n_] := recFib[n] =..." is a nice trick to give memory to a function, see "Functions That Remember Values They Have Found" in the documentation.)
A better way is implementing the general formula:
fib[n_] := With[{goldenRatio = (Sqrt[5] + 1)/2},
Simplify[1/Sqrt[5] (goldenRatio^n - (1 - goldenRatio)^n)]]
(Yes, I know, there is a constant "GoldenRatio" predefined, but when using that the simplification to integers does not work ... An idea anybody?) From this formula it is obvious that the term "(1 - goldenRatio)^n" vanishes for large n (its absolute value is smaller than 1). Consequently the expression
fibApprox[x_]:= GoldenRatio^x/Sqrt[5.]
serves as a good approximation - and one can see that the growth is exponential (as it should be with rabbits ...)
Ciao Henrik