Sum of first n Fibonacci numbers with respect to n?

• dimension10
In summary, The nth Fibonacci number is defined as \frac{{(1+\sqrt{5})}^{n}-{(1-\sqrt{5})}^{n}}{{2}^{n}\sqrt{5}} and the sum of the first n Fibonacci numbers can be written as \frac{{(1+\sqrt{5})}^{n+2}-{(1-\sqrt{5})}^{n+2}}{{2}^{n+2}\sqrt{5}}-1. This can be derived from the elegant formula F_0+F_1+...+F_n=F_{n+2}-1, using the correct formula for F_{n+2}.
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.

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

Isn't it just an obvious application of

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

definition?

micromass said:
Hi dimension10!

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

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

micromass said:
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

Thanks.

So we could write it as:

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

1. What is the formula for finding the sum of the first n Fibonacci numbers?

The formula for finding the sum of the first n Fibonacci numbers is (F(n+2) - 1), where F(n) is the nth Fibonacci number.

2. How do you calculate the sum of the first n Fibonacci numbers?

To calculate the sum of the first n Fibonacci numbers, you can use the formula (F(n+2) - 1), where F(n) is the nth Fibonacci number. Alternatively, you can also use a loop to add each Fibonacci number from 1 to n.

3. What is the significance of the sum of the first n Fibonacci numbers?

The sum of the first n Fibonacci numbers is significant because it can help in understanding the growth pattern of the Fibonacci sequence. It can also be used in various mathematical applications and problems.

4. Can you find the sum of the first n Fibonacci numbers if n is a large number?

Yes, the formula (F(n+2) - 1) can be used to find the sum of the first n Fibonacci numbers even if n is a large number. However, for very large values of n, it may be more efficient to use a loop or a more optimized algorithm.

5. What happens if n is equal to 0 or a negative number in the sum of the first n Fibonacci numbers?

If n is equal to 0 or a negative number, the sum of the first n Fibonacci numbers will be 0. This is because the formula (F(n+2) - 1) will result in a negative or 0 value for these inputs.

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