Fibonacci sequence

The Fibonacci sequence is a sequence of integers in which the first and second terms are both equal to 1 and each subsequent term is the sum of the two preceding it. The first few terms are $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...$ .

Recursion

The Fibonacci sequence can be written recursively as $F_1 = F_2 = 1$ and $F_n=F_{n-1}+F_{n-2}$ for $n \geq 3$ . This is the simplest nontrivial example of a linear recursion with constant coefficients. There is also an explicit formula below.

Readers should be wary: some authors give the Fibonacci sequence with the initial conditions $F_0 = F_1 = 1$ (or equivalently $F_1 = 1, F_2 = 2$ ). This change in indexing does not affect the actual numbers in the sequence, but it does change which member of the sequence is referred to by the symbol $F_n$ and so also changes the appearance of certain identities involving the Fibonacci numbers.

Running Backwards

As with many linear recursions, we can run the Fibonacci sequence backwards by solving its recursion relation for the term of smallest index, in this case $F_{n - 2} = F_{n} - F_{n - 1}$ . This allows us to compute, for example, that $F_0 = F_2 - F_1 = 0$ , $F_{-1} = 1$ , $F_{-2} = -2$ , and so on. Because these preceding terms are uniquely defined by the recursion, one frequently sees the definition of the Fibonacci sequence given in the form $F_0 = 0$ , $F_1 = 1$ and $F_n = F_{n - 1} + F_{n - 2}$ for $n \geq 2$ . In general, one can show that $F_n = (-1)^{n+1}F_{-n}$ .

$\phi$ and Binet's Formula

Main article: Binet's formula

The ratios $\frac{1}{1}$ , $\frac{2}{1}$ , $\frac{3}{2}$ , $\frac{5}{3}$ , $\frac{8}{5}$ , ..., between successive terms of the sequence tend towards the limit $\frac{1 + \sqrt{5}}{2}$ , a constant often denoted $\varphi$ (the Greek letter phi, also written $\phi$ ). $\varphi$ is a solution of the quadratic equation $x^2-x-1=0$ . The other root is $-\varphi^{-1} = \frac{1-\sqrt{5}}{2}$ . One possible explanation for this fact is that the Fibonacci numbers are given explicitly by Binet's formula. It is $F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt{5}}$ . (Note that this formula is valid for all integers $n$ .) It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre.

Identities

The most important identity regarding the Fibonacci sequence is its recursive definition, $F_{n+1} = F_n + F_{n-1}$ . The following identities involving the Fibonacci numbers can be proved by induction.

$F_0 + F_1 + \cdots + F_{n} = F_{n+2} - 1$
$F_0 - F_1 + F_2 - \cdots - F_{2n-1} + F_{2n} = F_{2n-1} - 1$
$F_0^2 + F_1^2 + F_2^2 + \cdots + F_n^2 = F_n \cdot F_{n+1}$
$F_{n-1}\cdot F_{n+1} = F_{n}^2 + (-1)^n$
$F_{m+n+1} = F_{m+1} \cdot F_{n+1} + F_{m} \cdot F_n$

Problems

Introductory

The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 }$
(2000 AMC 12, Problem 4)
A colony has rabbit. A rabbit produces one offspring every month. An offspring rabbit takes one month to grow up. Find a formula for the number of rabbits (including offspring) in the th month.
1. How about if the colony starts with $a$ rabbits and $b$ offspring?
2. Use this result to prove the identity $F_{m+n+1} = F_{m+1} \cdot F_{n+1} + F_{m} \cdot F_n$ .
Find $\gcd(F_n,F_{n+1})$ .
Prove the above identites.

Intermediate

Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle.
(Manhattan Mathematical Olympiad 2004)
Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length?
(1998 AIME, Problem 8)
A fair coin is to be tossed $10_{}^{}$ times. Let $i/j^{}_{}$ , in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$ .
(1990 AIME, Problem 9)
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$ .
(1998 AIME, Problem 13)

Olympiad

Determine the maximum value of $m^2 + n^2$ , where $m$ and $n$ are integers satisfying $m, n \in \{ 1,2, \ldots , 1981 \}$ and $( n^2 - mn - m^2 )^2 = 1$ .
(1981 IMO, Problem 3)