Fibonacci and related numbers

Overview of Fibonacci Sequence

  • The Fibonacci sequence is a specific type of sequence characterised by the fact that every number after the first two is the sum of the two preceding ones.

  • The first few terms are commonly initiated as 0, 1, but can also start with 1, 1 depending on the context.

Properties of Fibonacci Sequence

  • The ratio of consecutive terms in a Fibonacci sequence converges to an irrational number known as the golden ratio, denoted typically as φ. As n approaches infinity, this ratio tends towards approximately 1.61803.

  • Fibonacci numbers have interesting relationships with other sequences, such as Lucas numbers. A Lucas number is a number in a sequence defined similarly to the Fibonacci sequence.

  • Fibonacci numbers appear surprisingly often in nature and in art, architecture, and music. Furthermore, they have numerous applications in computer science, mathematics, and theoretical physics.

Generating the Fibonacci Sequence

  • A Fibonacci sequence can initially be generated by starting with 0 and 1 (or two 1s) then proceeding to add the last two terms together to generate the next term in the sequence.

  • Alternatively, the nth Fibonacci number can be computed directly using Binet’s formula: Fn = ((φ^n) - ((-φ)^(-n))) / sqrt(5). However, this is not normally used for computation, as it involves irrational numbers and the need for rounding.

Fibonacci Identities

  • There also exist numerous identities involving Fibonacci numbers such as Fn^2 + Fn^2 = F_(2n), which illustrates the relationship between the squares of Fibonacci numbers.

  • Cassini’s Identity: For any natural number n, where Fn represents the nth Fibonacci number, the following holds true: Fn^2 − Fn−1 × Fn+1 = (−1)^(n−1).

Use of Fibonacci Numbers in Problem Solving

  • Understanding the properties of the Fibonacci sequence is beneficial for solving problems in diverse areas of mathematics, including algebra, number theory, and particularly in advanced topics such as infinite series, matrices, and algorithms.

  • Quizzes and problems that involve identifying or working with Fibonacci numbers are common in mathematics. Familiarity with these numbers and their properties can help in quickly identifying them and in solving such problems.

Fibonacci Numbers and Recurrence Relations

  • The Fibonacci sequence is commonly represented through recurrence relations, defined as Fi = Fi-1 + Fi-2, with initial conditions F0 = 0 (or 1) and F1 = 1.

  • Recurrence relations are particularly useful in understanding and computing Fibonacci numbers, especially in computer algorithms.

  • Understanding recursive solutions is a fundamental aspect in the study of Fibonacci numbers and in other areas of mathematics and computer science.