Difference between Series and Sequence
Last Updated :
19 Mar, 2025
Sequence and Series are the most important topics in math, though many people get confused between them, they can easily be differentiated. Sequence refers to an arrangement in a particular order in which the related terms follow each other. When a sequence follows a particular pattern, it is called a progression. It is not the same as a series, which is defined as the summation of the sequence's elements.
Basic difference between Sequence and SeriesSequence
A sequence is an ordered list of numbers arranged according to a certain rule or pattern. Each number in a sequence is called a term.
The terms of a sequence are usually denoted as:
a1, a2, a3, …, an, …
The subscript indicates the position of each term within the sequence:
- First term = a1
- Second term = a2
- Third term = a3
The nth term is the number at the nth position of the sequence and is denoted by an. This term is also called the general term of the sequence.
For example, the sequence is 2, 4, 6, 8, 10, 12, . . .
- Here, 2 is the first term, 4 is the second term,6 is the third term, and so on.
- The dots at the end (. . .) indicate that the sequence continues indefinitely.
- This sequence has a constant difference (common difference) of 2, as each term is obtained by adding 2 to the previous term.
The sequence can be classified into different types:
- Arithmetic Sequence - An arithmetic sequence is defined as a sequence of numbers in which the difference between one term and the next term remains constant.
- Geometric Sequence - A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
- Harmonic Sequence - Harmonic Sequence is defined as a sequence of real numbers obtained by taking the reciprocals of an Arithmetic Progression that excludes 0.
- Fibonacci Sequence - The Fibonacci Sequence is a series of numbers starting with 0 and 1, where each succeeding number is the sum of the two preceding numbers.
Series
A series is defined as the sum of terms of a sequence, where the order of the terms typically matters. Series can be classified into finite and infinite, depending on whether the underlying sequence has a finite or infinite number of terms.
- A finite series has a definite number of terms and thus an end.
- An infinite series continues indefinitely without ending.
Example:
- Finite series: 1 + 3 + 5 + 7 + 9
- Infinite series: 1 + 3 + 5 + 7 + …
Different types of series include:
- Geometric Series - In a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease.
- Harmonic Series - Harmonic series is the inverse of an arithmetic progression. In general, the terms in a harmonic progression can be denoted as 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d) …. 1/(a + nd).
- Power Series - Power series is a type of infinite mathematical series that involves terms with a variable raised to increase powers an to infinite level.
- Alternating Series - The Alternating Series is a mathematical series where the sign of each term alternates between the positive and negative terms.
- Exponent Series - An exponential series is an infinite series representation of the exponential function, often used to express the function in a form that is easier to manipulate mathematically.
Sequence vs Series
This table comprises differences between sequence and series :
Aspect | Sequence | Series |
|---|
Definition | Sequence elements are placed in a particular order following a particular set of rules. | In series, the order of the elements is not necessary. |
|---|
Order | It is just a collection of elements in a particular pattern. | It is a sum of elements that follows a pattern. |
|---|
Notation | Represented as a1, a2, a3...... | Represented as Sn = a1 + a2 + a3 + a4.... |
|---|
Nature | The order of appearance of the number is important. | - Finite series sums are order-independent (due to commutativity).
- Infinite series can change sums based on term order.
|
|---|
Finite Example | Finite Sequence: 1, 2, 3, 4, 5 | Finite Series: 1 + 2 + 3 + 4 + 5 |
|---|
Infinite Example | Infinite Sequence: 1, 2, 3, 4....... | Infinite Series: 1 + 2 + 3 + 4 + 5..... |
|---|
Key Difference | A sequence is just a list of numbers. | A series is the summation of a sequence’s terms. |
|---|
Application | Used in computer science, physics, and patterns. | Used in calculus, economics, and physics for application. |
|---|
Related Articles,
Practice Questions on Difference between Series and Sequence
Question 1: Identify the sequence type: 3, 6, 12, 24, 48, …
Answer:
Geometric Sequence (Common ratio = 2).
Question 2: Determine if the sequence is arithmetic, geometric, or harmonic: 5, 9, 13, 17, 21, …
Answer:
Arithmetic Sequence (Common difference = 4).
Question 3: Classify the sequence: 1, 12, 13, 14, 15, …
Answer:
Harmonic Sequence (Reciprocals form an arithmetic sequence: 1, 2, 3, 4, 5, …).
Question 4: Identify the pattern: 0, 1, 1, 2, 3, 5, 8, …
Answer:
Fibonacci Sequence (Each term is the sum of the two preceding terms).
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