## Alternative Form For Fibonacci Sequence

The Fibonacci sequence 1, 1, 2, 3, 5, 8, 11... - is defined by the recurrence relation
$F_{n+1}=F_n+F_{n-1}$
.Br /> In particular,
$F_3=F_2+F_1$

$F_4=F_3+F_2$

$F_5=F_4+F_3$

$\vdots = \vdots+ \vdots$

$F_{k+1}=F_k+F_{k-1}$

$F_{k+1}+F_k+...+F_4+F_3=F_k+...+F_4+F_3+F_2+F_{k-1}+...+F_3+F_2+F_1$
$F_{k+1}=F_{k-1}+F_{k-2}F_{k-3}...+F_3+2F_2+F_1$
$F_2=1$
$F_{k+1}=F_{k-1}+F_{k-2}F_{k-3}...+F_3+F_2+F_1+1$