The Fibonacci Sequence

The Fibonacci sequence is a well-known sequence, named after the Italian mathematician Leonardo Fibonacci. The first two terms of the sequence are both 1, and subsequent terms in the sequence are generated by adding the two preceding numbers.

Starting from the third term onward, the sequence can be represented using a rule. & F_1 = F_2 = 1 & F_n = F_(n - 1) + F_(n - 2) This is an example of recursive rule as the nth term depends on the values of the previous two terms. This indicates that to find, for instance, the seventh term F_7 of the sequence, it is necessary to have information about the fifth and sixth terms.

Sometimes, the first two numbers are defined as F_1=0 and F_2=1. This, however, does not change the sequence in any way other than by increasing the index of every term by one.

Extra

The Fibonacci Sequence and the Golden Ratio

Consider the ratio between two consecutive terms of the sequence. Pay close attention to the value of the ratio as n increases.

The ratio of two consecutive Fibonacci numbers approaches the golden ratio φ, which is approximately 1.618034. φ= 1+sqrt(5)/2 ≈ 1.618034 Even more surprising fact is that any Fibonacci number can be calculated using the golden ratio. F_n = φ^n - (1- φ)^n/sqrt(5)