Fibonacci Sequence

About fifteen years ago, a student was talking about the Fibonacci Sequence, and told me that the Head of Mathematics in his school had stated that there was no general formula for the nth term of this Sequence. So I set myself the task of finding such a formula.

To my surprise, when I was reading a book this year by Professor Ian Stewart called "Does God Play Dice?", I found him giving an incorrect (though approximate) formula. So it is possible that the right formula is not generally known.

The sequence itself, 1, 1, 2, 3, 5, 8, 13 and so on, is very easily calculated. Each term is found by adding together the previous two terms. But finding a formula is quite another matter. The formula for the nth term is: Ar^(n-1) + Bs^(n-1), where: A = (5+\5)/10, B = (5-\5)/10, r = (1+\5)/2, and s = (1-\5)/2 With limited keys at my disposal, I am having to use \5 to mean the square root of 5.

By using different values for the coefficients A and B, other series of the same sort may be generated; but the Fibonacci Sequence is the best known, thanks in part to Dan Brown`s book, "The da Vinci Code".

I would be happy to enter into correspondence with anyone who would like to know how this formula is derived.