# 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

Staff Emeritus
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:

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