## Formula for Sum of Fibonacci Terms

The Fibonacci sequence is defined by
$F_1=F_2=1, F_{n+1}=F_{n-1}+F_n, n \ge 2$
.
To derive the identity
$1+1+2+3+...+F_n=F_{n+2}-1$
start by adding the system of equations
\begin{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}

All the terms on the right cancel apart from
$F_1$
and
$F_{n+2}$
giving
$1+1+2+3+...+F_n=F_{n+2}-1$
.