\[F_1=F_2=1, F_{n+1}=F_{n-1}+F_n, n \ge 2\]
.Theorem
\[F_2+F_4+...+F_{2n}=F_{2n+1}-1\]
. (1)Proof
Use this Formula for Sum of Fibonacci Terms and Formula for Sum of Odd Fibonacci Terms.
\[F_1+F_2+...+F_{2n}=F_{2n+2}-1\]
\[F_1+F_3+...+F_{2n-1}=F_{2n}\]
Subtracting these identities gives
\[\begin{equation} \begin{aligned} F_2+F_4+...+F_{2n} &= F_{2n+2}-F_{2n}-1 \\ &= F_{2n+1}+F_{2n}-F_{2n}-1 \\ &= F_{2n+1}-1 \end{aligned} \end{equation}\]