Let $F_{i}$ be the $i^{th}$ Fibonacci number
(a) Write $F_{1}-F_{2}+F_{3}-F_{4}+...+F_{2n-1}-F_{2n}$ as a summation
(b) Prove $F_{1}-F_{2}+F_{3}-F_{4}+...+F_{2n-1}-F_{2n}=1-F_{2n-1}$
I'm quite confused. I've looked at how to express the Fibonacci sequence as a summation, as well as how to express just the odd or even numbers as a summation and based on that, this is my current understanding of the question.
(A) $\sum_{n=1}^{2n} f(n)=\sum_{n=1}^{2n} (f(n)-f(n+1))$
(B) Hoping that I can add to this as we proceed along to part A?
Is that right or have I understood something wrong here?
For Part (b), note that $$F_{2k-1}-F_{2k}=F_{2k-1}-\big(F_{2k-1}+F_{2k-2}\big)=-F_{2k-2}=F_{2k-3}-F_{2k-1}$$ for all $k=1,2,3,\ldots$ (noting that $F_{-1}=1$).