The formula for finding the nth Fibonacci number is?

Question

The formula for finding the nth Fibonacci number is:

F(n) = (φ^n - (1-φ)^n) / √ 5

I dont understand what the swiggily thing is? (Where is it on my calculator)What does f and n stand for? Is there an easier formula that I can understand?

Answer 1

F(n) means the nth Fibonacci number. So F(4) = 3, as the sequence begins
1,1,2,3,5,8,13, ...

The next one F(8) = 21

The squiggly thing (the Greek letter phi) represents
(1+√ 5 )/2, and so

1-φ = (1-√ 5)/2

As far as I know, there isn't an easier formula. Not many people know this one.

Answer 2

The swiggly thing is the greek letter phi, which stands for the golden ratio, or something like that. You can derive this formula as follows.

The Fibonacci numbers are related by the formula:

F(n+2) = F(n+1) + F(n)

or rearranging

F(n+2) - F(n+1) - F(n) = 0.

Now notice that if x is a number which satisfies

x^2 - x -1 = 0,

then

x^{n+2} - x^{n+1} - x^n = x^n ( x^2 -x -1) = 0.

Thus F(n) = x^n will satisfy the relation defining the Fibonacci numbers. Therefore we should find the solutions of the equation x^2 - x - 1 = 0. Using the quadratic formula the two solutions are

x = (1 +/- sqrt{5})/2

Now, when you take the "+" that is the swiggly thing, phi, or the golden ratio. Notice that

1 - (1+ sqrt{5})/2 = (1 - sqrt{5})/2, and the two solutions of x^2 - x -1 = 0 are phi and 1-phi. Thus a sequence of numbers defined by the relation S(n+2) = S(n+1) + S(n) will be given by the formula

S(n) = a*phi^n + b*(1-phi)^n

for some constants a and b. If S(n) is the Fibonacci sequence then S(1) = S(2) = 1. Thus

1 = a*(1+sqrt{5})/2 + b*(1-sqrt{5})/2 = a*(1+sqrt{5})^2/4 + b*(1-sqrt{5})^2/4

I won't write it all out here, but if you use these equations to solve for a and b, you should get a = 1/sqrt{5} and b = -1/sqrt{5}.

Answer 3

F(n) is just the usual mathematical shorthand for "the Fibonacci number which is 'n' places along in the series". That formula in your question is the only one that will give the answer in a single step.

An easier formula is:
F(2n-1) = F(n-1)^2 + F(n)^2
F(2n) = (2F(n-1) + F(n)) * F(n)

So for example knowing that 8 and 13 are consecutive Fibonacci numbers, we can construct 8^2 + 13^2 = 64 + 169 = 233, and (2*8 + 13) * 13 = 29 * 13 = 377, and know that 233 and 377 are two more adjacent Fibonacci numbers twice as far up the series as 8 and 13 were. From those two, we would know that 377 and 610 are the next two. This is the formula that a computer would use, over and over again, to get pretty quickly to any Fibonacci number that you asked for.

Answer 4

I think you will have a very, very hard time finding a closed formula for this sequence. Since this is work-related, I'm sure you didn't just get this problem to solve. Sometimes it helps if you can provide more background on what you really need, there could be an alternative way to formulate a solution. Perhaps you could outline your actual problem and show how you came to this sequence as the solution. Alternatively, this problem may be only part of your actual problem and if you describe the whole problem there might be another way.

Answer 5

It is not on your calculator, it is the number phi. You can find out about it on the link below (or in your textbook).

http://en.wikipedia.org/wiki/Phi