\[F_1=F_2=1, F_{n+1}=F_{n-1}+F_n, n \ge 2\]
.To derive the identity
\[1+1+2+3+...+F_n=F_{n+2}-1\]
start by adding the system of equations\[\begin{equation} \begin{aligned} & F_3=F_2-F_1 \\ & F_4=F_3-F_2 \\ & F_5=F_4-F_3 \\ & \vdots = \vdots \vdots \\ & F_n=F_{n+2}-F_{n+1} \end{aligned} \end{equation}\]
All the terms on the right cancel apart from
\[F_1\]
and \[F_{n+2}\]
giving \[1+1+2+3+...+F_n=F_{n+2}-1\]
.