\[F_{n+1}=F_n+F_{n-1}\]
.Br />
In particular,\[F_3=F_2+F_1\]
\[F_4=F_3+F_2\]
\[F_5=F_4+F_3\]
\[\vdots = \vdots+ \vdots\]
\[F_{k+1}=F_k+F_{k-1}\]
Adding these up gives
\[F_{k+1}+F_k+...+F_4+F_3=F_k+...+F_4+F_3+F_2+F_{k-1}+...+F_3+F_2+F_1\]
Many terms cancel, leaving
\[F_{k+1}=F_{k-1}+F_{k-2}F_{k-3}...+F_3+2F_2+F_1\]
\[F_2=1\]
so we can write\[F_{k+1}=F_{k-1}+F_{k-2}F_{k-3}...+F_3+F_2+F_1+1\]