I am currently working through the various properties of the binomial coefficient and was up to the identity
$\sum_{k=0}^n\binom{n-k}{k} = F_{n+1}$
The proof is provided elsewhere on this website, here specifically. I have reproduced the user Adi Dani's proof below.
$$\begin{align} F_{n+1}&=\sum_{k=0}^n\binom{n-k}{k}\\ &=\sum_{k=0}^{n}\Bigg(\binom{n-k-1}{k}+\binom{n-k-1}{k-1}\Bigg)\\ &=\sum_{k=0}^{n-1}\Bigg(\binom{(n-1)-k}{k}+\binom{n-2-(k-1)}{(k-1)}\Bigg)\\ &=\sum_{k=0}^{n-1}\binom{(n-1)-k}{k}+\sum_{k=0}^{n-1}\binom{n-2-(k-1)}{k-1}\\ &=\sum_{k=0}^{n-1}\binom{(n-1)-k}{k}+\sum_{j=0}^{n-2}\binom{(n-2)-j}{j}\\ &=F_{n}+F_{n-1} \end{align}$$
It's all clear to me except for the changing of the index on the summation. How are you able to reduce a summation by one or two members in the series without consequence?
As an addendum to this question, I've noted that the only proofs I've had trouble with so far have involved changing the index on a summation. As such, if anyone had some good resources for looking at summations in more detail (especially if there are exercises) then that would be massively appreciated.