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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?
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.
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.
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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