A sequence in mathematics is an arrangement of numbers in a sequential manner that follows a specific rule; each term of a sequence is related to its previous and successive term by that given rule.

More formally, a sequence may be defined as a mapping or a function f: 𝐍→X defined as f(n) = xn for every n in 𝐍

such that all the xn; n = 1, 2, 3,… is a sequence in X governed by the rule f.

**Some examples of a sequence are:**

- Sequence 2, 6, 12, 30,… the general rule for this particular sequence is n(n + 1); n = 1, 2, 3,…
- 1, 1/2, 1/3, 1/4,… whose general rule is 1/n where n is a Natural Number.

## Recursive Sequence

A recursive sequence is a sequence formed by a recursive function. Recursion is a process in which a sequence is formed by selecting an initial term to begin the sequence and repeatedly using the previous term to find the next term. Thus, recursion is a **recursive function** that uses the initial or preceding values to get the successive terms. There are two steps of a recursion:

**Basic Step: **Specifies a collection of starting values or initial values of the function.

**Recursive Step: **Gives the rule to form new elements based on known values or previous values of the sequence.

Sometimes an exclusion set is also defined, which specifies a set of values that are not to be included in the recursion process.

Example of a recursive sequence:

The sequence of natural numbers.

**Basic Step: **Let f be the recursive function whose initial value f_{o} = 0

**Recursive Step: **f_{n} = f_{n} + 1; According to recursive rule

f_{1} = f_{o }+ 1 = 0 + 1

f_{2} = f_{1} + 1 = 2 and so on we get the sequence of natural numbers

## Fibonacci Sequence

A prominent example of a recursive sequence is a Fibonacci Sequence. This sequence is proven to be one of the most intriguing and ubiquitous of all number sequences in mathematics. **Fibonacci Sequence,** also known as Fibonacci Numbers, is defined recursively as:

For n be any number, n ≥ 0, if F_{n } is the nth Fibonacci number, then we have

**Basic Step: **The initial values of the recursion, F_{o}= 0 and F_{1}= 1

**Recursive Step: **The recurrence relation is defined as

F_{n} = F_{n – 1} + F_{n – 2} , n ≥ 2

The successive Fibonacci Numbers are found by adding the preceding two numbers of the sequence. Hence, the terms in the sequence are 0, 1, 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, that is, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on.

### Some Properties of Fibonacci Numbers

The number pattern has some interesting properties, which makes it such a significant number sequence.

- Successive numbers of the sequence have a recurrence relation.
- The Greatest Common Divisor of successive terms of the sequence is 1.

For n ≥ 0 and F_{n} be recursion function for the sequence, gcd( F_{n}, F_{n+1}) = 1. Let us check this property b taking an example, gcd(F_{5}, F_{6}) = gcd(5, 8) = 1 and gcd(F_{9}, F_{10}) = gcd(34, 55), now factors of 34 are 1, 2, 17 and 34 and factors of 55 are 1, 5, 11 and 55. Thus, gcd(34, 55) = 1.

- For n ≥ 0, gcd(F
_{n}, F_{n + 2}) = 1

We can also check this one by taking an example, F_{5} = 5 and F_{5 + 2} = F_{7} = 13, clearly

gcd(5, 13) = 1.

- The sum of any six consecutive Fibonacci Numbers is a multiple of 4.

Let us take, F_{2} + F_{3} + F_{4} + F_{5} + F_{6} + F_{7} = 1 + 2 + 3 + 5 + 8 + 13 = 32 = 4 × 8.

- The ratio of any two consecutive Fibonacci Numbers is approximately equal to the Golden Ratio.

The value of the Golden ratio, φ = 1.618, approximately

F_{6} : F_{5} = 8 : 5 = 1.6 which is close to the Golden Ratio.