Sum of first n Fibonacci numbers with respect to n?

1. Jun 23, 2011

dimension10

I know that the nth Fibonacci number is defined as:

$$\frac{{1+\sqrt{5}}^{n}-{1-\sqrt{5}}^{n}}{{2}^{n}\sqrt{5}}$$

But may I know the formula for the sum of the first n Fibonacci numbers with respect to n? Thanks.

2. Jun 23, 2011

micromass

Hi dimension10!

That formula you give can't possibly be correct, since it evaluates to 0... Did you forget to add some brackets?

Anyway, the most elegant formula for the sum of the first n Fibonacci numbers is

$$F_0+F_1+...+F_n=F_{n+2}-1$$

Using the (correct) formula for $F_{n+2}$ gives you the desired formula.

Check http://en.wikipedia.org/wiki/Fibonacci_number

3. Jun 23, 2011

Staff: Mentor

Isn't it just an obvious application of

$$F_n = F_{n-1} + F_{n-2}$$

definition?

4. Jun 23, 2011

dimension10

Yes.I meant
$$\frac{{(1+\sqrt{5})}^{n}-{(1-\sqrt{5})}^{n}}{{2}^{n}\sqrt{5}}$$

Thanks.

5. Jun 23, 2011

dimension10

So we could write it as:

$$\frac{{(1+\sqrt{5})}^{n+2}-{(1-\sqrt{5})}^{n+2}}{{2}^{n+2}\sqrt{5}}-1$$