Alternating Sum of Fibonacci Numbers: Get (-1)^(n+1) Factor

[Bellarosa]

1.Obtain the following formula for the alternating sum of Fibonacci Numbers

f(1)-f(2)+f(3)-f(4)+...+(-1)^(n+1) f(n)= (-1)^(n+1) [f(n-1) ]+1

Homework Equations

3. I subtracted the formulas for the sum of the first n Fibonacci numbers with odd indices together with the first n Fibonacci numbers with even indices

f(1)+f(3)+f(5)+...+f(2n-1) = f(2n)
- [f(2)+f(4)+f(6)+...+f(2n) = [f(2n+1) ]- 1]

= f(1)-f(2)+f(3)-f(4)+f(5)-f(6)+...+f(2n-1)-f(2n) = f(2n) - [f(2n+1)] +1

what I am having trouble with is obtaining the (-1)^(n+1) factor
I am thinking that if I factor the first two terms of the difference I would get the (-1) but as for its exponent I am not quite clear on how to derive it

 

[mighty2000]

.[/b]



Thank you for your question regarding the alternating sum of Fibonacci numbers. I would be happy to assist you in obtaining the formula that you are looking for.

First, let's start with the definition of the Fibonacci sequence. The sequence is defined as follows:
f(1) = 1
f(2) = 1
f(n) = f(n-1) + f(n-2) for n > 2

Now, let's look at the sum of the first n Fibonacci numbers with odd indices and the sum of the first n Fibonacci numbers with even indices. We can write these sums as follows:
f(1) + f(3) + f(5) + ... + f(2n-1) = f(2n)
f(2) + f(4) + f(6) + ... + f(2n) = f(2n+1) - 1

Subtracting these two equations, we get:
f(1) - f(2) + f(3) - f(4) + ... + f(2n-1) - f(2n) = f(2n) - (f(2n+1) - 1)
= -f(2n+1) + f(2n) + 1

Now, let's look at the (-1)^(n+1) factor. This factor is present because we are alternating between adding and subtracting terms in the sum. When n is odd, we have (-1)^(n+1) = -1, and when n is even, we have (-1)^(n+1) = 1. This is why we see the alternating sign in the sum.

To obtain the formula that you are looking for, we just need to substitute f(2n+1) with f(n) in the previous equation. This gives us:
f(1) - f(2) + f(3) - f(4) + ... + (-1)^(n+1) f(n) = -f(n) + f(2n) + 1
= (-1)^(n+1) (f(n-1) + 1) + 1
= (-1)^(n+1) [f(n-1) + 1] + 1

I hope this helps you