Sequences

Sequences is an ordered list of numbers that follow a particular pattern. We see sequences in many places in real life. For example, the house numbers in a row, page numbers of a book.

Definition

A sequence is a list of numbers (or elements) that exhibits a particular pattern. Each element in a sequence is called a term. A sequence can be finite , meaning it has a specific number of terms, or infinite meaning it continues indefinitely.

We denote the terms of a sequence by $a_{1},a_{2},a_{3},a_{4}.......a_{n}$.The subscript denotes the position of the term. The $n^{th}$ term is the number at the $n^{th}$ position of the sequence and is denoted by $a_{n}$.The $n^{th}$ term is also called the general term of the sequence.

Example

Let us understand this with an example. $1,3,5,7,9,11......$ is a sequence where there is a common difference of $2$ between any two terms and the sequence goes on increasing up to infinity unless the upper limit is given. This example is for the infinite sequence.

In this example $a_{1}=1,a_{2}=3,a_{3}=5,a_{4}=7,a_{5}=9,a_{6}=11$ .

Another example $1,2,4,8,16,32,64$ is a sequence where there is a common ratio of $2$ between any two terms and the sequence ends at $64$. This example is for the finite sequence.

In this example $a_{1}=1,a_{2}=2,a_{3}=4,a_{4}=8,a_{5}=16,a_{6}=32,a_{7}=64$ .

Types of Sequences

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each successive term is a sum of its preceding term and a fixed number. This fixed number is called a common difference. The terms of the arithmetic sequence are of the form $a,a+d,a+2d,.......$

Example: $1,4,7,10,.......$ Here $a=1,d=3$

Geometric Sequence

A geometric sequence is a sequence where every term bears a constant ratio to its preceding term. This ratio is called the "common ratio". The terms of the geometric sequence are of the form $a,ar,ar^{2},......$

Example: $1,4,16,64,.......$ Here $a=1,d=4$

Fibonacci Sequence

Fibonacci sequence is a sequence where every term is the sum of the last two preceding terms.

Example:

$0,1,1,2,3,5,8,13,21,34$