Mathematics · Ch 5 — Binomial Theorem, Sequences and Series

Fibonacci Sequence

5.6.1

Fibonacci Sequence

The Fibonacci sequence is a sequence where every term from the third onward is the sum of the two terms before it. Starting from $1,1$ :

1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ \ldots \qquad x_n=x_{n-1}+x_{n-2}\ (n\ge3),\ \ x_0=x_1=1.

Named after Fibonacci (Leonardo of Pisa / Leonardo Pisano), who introduced these numbers in his 1202 book Liber Abaci. The son of a Pisan merchant, Fibonacci travelled and traded widely across North Africa, where he learned the Hindu-Arabic numeral system — at the time unknown to the Latin-speaking world — and his book helped introduce Europe to the decimal system and the concept of zero, alongside geometry, commercial arithmetic and irrational numbers.

n=1,2,3,\ldots,14,\ldots \qquad x_n=1,1,2,3,5,8,13,21,34,55,89,144,233,377,\ldots

(For example, the $8^{th}$ term is the sum of the $6^{th}$ and $7^{th}$ : $x_8=8+13=21$ .) …

Figure 5.4Fibonacci branching in a plant

What this figure shows. A schematic flowering stem where each branch splits following the Fibonacci count at successive levels — the classic illustration of Fibonacci numbers appearing in natural branching/leaf patterns. …

Figure 5.5Fibonacci numbers inside Pascal's triangle

What this figure shows. Pascal's triangle drawn as rows of circles with diagonal lines drawn across each rising diagonal; summing the circled entries along each red diagonal reproduces the Fibonacci sequence, visually linking $^nC_r$ to $x_n$ . …

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