\begin{align*} F_0 + F_2 + \cdots + F_{2k} + F_{2k+2} ~ \amp = F_{2k+1} - 1 + F_{2k+2} \amp \text{by ind. hyp.}\\ \amp = F_{2k+1} + F_{2k+2} - 1 \amp\\ \amp = F_{2k+3} - 1 \amp \text{by recursive def.} \end{align*}