Fibonacci Numbers

Lesson · Intermediate

Algebra

The Fibonacci sequence is defined by

F_0=0,\qquad F_1=1

and

F_{n+2}=F_{n+1}+F_n

for n ≥ 0.

Thus,

0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ 89,\ 144,\ldots

Each term is the sum of the previous two terms.

$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$F_n$	$0$	$1$	$1$	$2$	$3$	$5$	$8$	$13$	$21$	$34$

$n$	$10$	$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$
$F_n$	$55$	$89$	$144$	$233$	$377$	$610$	$987$	$1597$	$2584$	$4181$

One of the most useful Fibonacci identities is

F_{m+n}=F_{m-1}F_n+F_mF_{n+1}\qquad(m\ge1,\ n\ge0)