(1)

\begin{align*} \sum_{k=0}^{n}F_{k} & =F_{0}+F_{1}+\sum_{k=2}^{n}F_{k}\\ & =F_{0}+F_{1}+\sum_{k=2}^{n}\left(F_{k-1}+F_{k-2}\right)\\ & =F_{0}+F_{1}+\sum_{k=1}^{n-1}F_{k}+\sum_{k=0}^{n-2}F_{k}\\ & =F_{0}+F_{1}+\LHS-F_{0}-F_{n}+\LHS-F_{n-1}-F_{n}\\ & =F_{1}+2\LHS-F_{n}-F_{n+1}\\ & =F_{1}+2\LHS-F_{n+2}\\ & =F_{n+2}-F_{1}\\ & =F_{n+2}-1 \end{align*}

(2)

\begin{align*} \sum_{k=1}^{n}F_{2k-1} & =F_{1}+\sum_{k=2}^{n}F_{2k-1}\\ & =F_{1}+\sum_{k=2}^{n}\left(F_{2k-2}+F_{2k-3}\right)\\ & =F_{1}+\sum_{k=1}^{n-1}\left(F_{2k}+F_{2k-1}\right)\\ & =F_{1}+\sum_{k=1}^{2\left(n-1\right)}F_{k}\\ & =F_{1}+\sum_{k=1}^{2\left(n-1\right)}F_{k}\\ & =F_{1}+F_{2\left(n-1\right)+2}-F_{1}-F_{0}\\ & =F_{2n}-F_{0}\\ & =F_{2n} \end{align*}

(3)

\begin{align*} \sum_{k=0}^{n}F_{2k} & =F_{0}+\sum_{k=1}^{n}F_{2k}\\ & =F_{0}+\sum_{k=1}^{n}\left(F_{2k-1}+F_{2k-2}\right)\\ & =F_{0}+\sum_{k=0}^{n-1}\left(F_{2k+1}+F_{2k}\right)\\ & =F_{0}+\sum_{k=0}^{2n-1}F_{k}\\ & =F_{2n+1}-F_{1}+F_{0}\\ & =F_{2n+1}-1 \end{align*}

(4)

\begin{align*} \sum_{k=0}^{n}F_{k}^{2} & =F_{0}^{2}+\sum_{k=1}^{n}F_{k}^{2}\\ & =F_{0}^{2}+\sum_{k=1}^{n}F_{k}\left(F_{k+1}-F_{k-1}\right)\\ & =F_{0}^{2}+\sum_{k=1}^{n}\left(F_{k}F_{k+1}-F_{k-1}F_{k}\right)\\ & =F_{n}F_{n+1}-F_{0}F_{1}+F_{0}^{2}\\ & =F_{n}F_{n+1} \end{align*}