# Sequences

A sequence is a set of numbers arranged one after another.

**a _{1}, a_{2}, a_{3} ,..., a_{n}**

3, 6, 9,..., 3n

The numbers **a _{1}, a_{2} , a_{3} , ...,** are called terms or elements of the sequence.

The subscript is the set of positive integers 1, 2, 3, ... The subscript indicates the place that a term occupies in the sequence.

The nth term is denoted by **a _{n}**.

### Calculation of a Sequence

#### By the Nth Term

**a _{n }**is a criterion that allows us to calculate any term of the sequence.

#### Example

**a _{n}= 2n − 1**

a_{1} = 2 ·1 − 1 = 1

a_{2} = 2 ·2 − 1 = 3

a_{3}= 2 ·3 − 1 = 5

a_{4} = 2 ·4 − 1 = 7

**1, 3, 5, 7, ..., 2n −1**

Not all sequences have a general term. For example, the sequence of prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23,...

#### By a Recursive Formula

A term is obtained by operating with the previous terms.

#### Example

Write a sequence whose first term is 2, knowing that each term is the square of the previous term.

**2, 4, 16, ... **

**Fibonacci Sequence** or Fibonacci Number

**1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...**

The first two terms are one and the other terms are obtained by adding the two previous terms.