# Difference Between Sequence and Series

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Sequence and Series: Mathematics plays a crucial role in the SSC and Railway Exams. Today, we will cover one of the important topics of the arithmetic section of the maths syllabus: “Sequence and Series”. We will cover the definition, Sequence and Series examples, and also the difference between Sequence, Series, and Progression. Let us start

Table of content

## Sequence and Series

What is Sequence?

Sequences are a set of numbers that are structured or defined according to a given rule, implying that they have something in common.

If the terms in a sequence are denoted by a1, a2, a3, a4,….etc., then the position of the term is denoted by 1,2,3,4,…..

A sequence can be defined by the number of terms in it, which can be finite or infinite.

For example: {2, 4, 8, 16, 32, 64, 128} is a sequence. Here, the rule is: Next number is the double of the preceding number

What is a Series?

series is succession of numbers joined together by signs.
The term ‘series’ is supposed to indicate the total of the numbers rather than the amount itself.

If a1, a2, a3, a4, ……. is a sequence, then the corresponding series is given by

SN = a1+a2+a3 + .. + aN

Note:  The series is finite or infinite depends on the fact that whether the sequence is finite or infinite.

For instance, the series associated with the first six odd numbers is 3+5+7+9+11+13.

How do sequences differ from sets?

Sequences are quite similar to sets, but the main distinction is that individual words in a sequence can recur repeatedly in different locations. A sequence’s length is equal to the number of phrases and might be finite or infinite.

What is Progression and how do they differ from the sequence?

Progressions are sets of numbers that are arranged according to some definite rule. The difference between a progression and a sequence is that a progression has a specific formula to calculate its nth term, whereas a sequence can be based on a logical rule like ‘a group of prime numbers.

For example: {12, 14, 16, 18, 20} is a progression. Its nth term is 2n.

## Types of Sequence and Series

Some of the most common examples of sequences are:

### Arithmetic Sequences

As per the NCERT definition, arithmetic Sequence is a sequence in which each term except the first is obtained by adding a fixed number (positive or negative) to the preceding term.
Thus any sequence a1, a2, a3… an, … is called an arithmetic progression if an + 1= an+ d, n ∈ N, where d is called the common difference of the A.P., usually we denote the first term of an A.P by a and the last term by l

### Geometric Sequences

A geometric sequence is one in which each term is formed by multiplying or dividing a definite number by the preceding number.

### Harmonic Sequences

A number series is said to be in harmonic sequence if the reciprocals of all its elements form an arithmetic sequence.

### Fibonacci Numbers

Fibonacci numbers create an intriguing number sequence in which each element is produced by adding two preceding elements and the sequence begins with 0 and 1. F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2 are the sequence definitions.

## Sequence and Series Formulas

The table given below mentions some basic arithmetic progression formulae

 Arithmetic Progression Sequence a, a+d, a+2d,……,a+(n-1)d,…. Common Difference or Ratio Successive term – Preceding term Common difference = d = a2 – a1 General Term (nth Term) an = a + (n-1)d nth term from the last term an = l – (n-1)d Sum of first n terms sn = n/2(2a + (n-1)d)

The table given below mentions some basic geometric progression formulae

 Geometric Progression Sequence a, ar, ar2,….,ar(n-1),… Common Difference or Ratio Successive term/Preceding term Common ratio = r = ar(n-1)/ar(n-2) General Term (nth Term) an = ar(n-1) nth term from the last term an = 1/r(n-1) Sum of first n terms sn = a(1 – rn)/(1 – r) if r < 1 sn = a(rn -1)/(r – 1) if r > 1

*Here:

• ‘a’ means the first term,
• ‘d’ means the common difference,
• ‘r’ means the common ratio,
• ‘n’ means the position of term,
• ‘l’ means the last term

## Difference Between Sequences and Series

A sequence is an itemised collection of components that allows for repeats of any sort, whereas a series is the sum of all parts. An arithmetic progression is a popular example of a sequence and series.

Let us find out how a sequence can be differentiated from a series.

 Sequences Series A sequence is a set of elements that follow a pattern A series is a sum of elements of the sequence The order of elements in the sequence is important The order of elements is not so important in the case of series Example of finite sequence: 1,2,3,4,5 Example of finite series: 1+2+3+4+5 Example of infinite sequence: 1,2,3,4,…… Example of infinite Series: 1+2+3+4+……

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