Fibonacci Sequence Properties
Important properties of Fibonacci Sequence are:
- We can easily calculate the Fibonacci Numbers using the Binet Formula:
Fn = (Φn – (1-Φ)n)/√5
where Φ is called Golden Ratio and its value is, Φ ≈ 1.618034.
Using this formula we can easily calculate the nth term of the Fibonacci sequence as, for
F34 = (Φ4 – (1-Φ)4)/√5 = ({1.618034}4– (1-1.618034)4)/√5 = 3
- For larger terms the ratio of two consecutive terms of the Fibonacci Sequence converges to the Golden Ratio.
This can be understood by the table added below,
Thus, it is evident that as the number becomes larger their ratio converges close to the Golden Ratio (1.618034).
- Multiplying a term of Fibonacci Sequence with Golden Ratio gives the next term of the Fibonacci sequence as,
F7 in Fibonacci Sequence is 13 then F8 is calculated as,
F8 = F7(1.618034) = 13(1.618034) = 21.0344 = 21 (approx.)
Thus, the F8 in the Fibonacci Sequence is 21.
- We can also calculate the Fibonacci Sequence for below zero numbers as,
F-n = (-1)n+1Fn
For example, F-2 = (-1)2+1F2 = -1
- Fibonacci Numbers are used to define other mathematical concepts such as Pascal Triangle and Lucas Number.
Fibonacci Sequence: Definition, Formula, List and Examples
Fibonacci sequence is a series of numbers where each number is the sum of the two numbers that come before it. The numbers in the Fibonacci sequence are known as Fibonacci numbers and are usually represented by the symbol Fₙ. Fibonacci sequence numbers start with the following 14 integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.
Fibonacci Sequence
Table of Content
- Fibonacci Sequence
- Fibonacci Sequence Formula
- Fibonacci Spiral
- Golden Ratio to find Fibonacci Sequence
- Golden Ratio Formula
- Fibonacci Series in Pascal’s Triangle
- Fibonacci Sequence Properties
- Fibonacci Sequence Examples
- Practice Problems on Fibonacci Sequence
- Fibonacci Sequence – FAQs
There are various applications of Fibonacci sequence in real life, such as in the growth of trees. As the tree grows, the trunk grows and spirals outward. The branches also follow the Fibonacci sequence, starting with one trunk that splits into two, then one of those branches splits into two, and so on.
Let’s learn about Fibonacci Sequence in detail, including Fibonacci sequence formula, properties, and examples.