Difference Between Sequence and Series

Sequence and Series is one of the important topics in Mathematics. Though many students tend to get confused between the two, these two can be easily differentiated. Sequence and series can be differentiated, in which the order of sequence always matter in the sequence but it’s not the case with series.

Sequence and series are the two important topics which deal with the listing of elements. It is used in the recognition of patterns, for example, identifying the pattern of prime numbers, and solving puzzles, and so on. Also, the series plays an important role in the differential equations and in the analysis process. In this article, let us discuss the key difference between the sequence and series in detail. Before that, we will see the brief definition of the sequence and series.

Definition of Sequence and Series in Maths

Sequence: The sequence is defined as the list of numbers which are arranged in a specific pattern. Each number in the sequence is considered a term. For example, 5, 10, 15, 20, 25, … is a sequence. The three dots at the end of the sequence represents that the pattern will continue further. Here, 5 is the first term, 10 is the second term,15 is the third term and so on. Each term in the sequence can have a common different, and the pattern will continue with the common difference. In the example given above, the common difference is 5. The sequence can be classified into different types, such as:

  • Arithmetic Sequence
  • Geometric Sequence
  • harmonic Sequence
  • Fibonacci Sequence

Series: The series is defined as the sum of the sequence where the order of elements does not matter. It means that the series is defined as the list of numbers with the addition symbol in between. The series can be classified as a finite series and infinite series which depends on the types of sequence whether it is finite or infinite. Note that, the finite series is a series where the list of numbers has an ending, whereas the infinite series is never-ending. For example, 1+3+5+7+.. is a series. The different types of series are:

  • Geometric series
  • Harmonic series
  • Power series
  • Alternating series
  • Exponent series (P-series)

What is the Difference Between Sequence and Series?

Here, the list of major differences between the sequence and series are given below:

Sequence Series
Sequence relates to the organization of terms in a particular order (i.e. related terms follow each other) and series is the summation of the elements of a sequence. Series can also be classified as a finite and infinite series.
In sequence, the ordering of elements is the most important In series, the order of elements does not matter
The elements in the sequence follow a specific pattern The series is the sum of elements in the sequence
Example: 1, 2, 4, 6, 8, . . . . n are said to be in a Sequence and 1 + 2 + 4 + 6 + 8 . . . . n is said to be in a series. A finite series can be represented as m1 + m2 + m3 + m4 + m5 + m6 + . . . . . + mn
General Form: \([p_{i}]_{i=1}^{n}\). Unending sequence like p1, p2, p3, p4, p5, p6, . . . . , pn, . . . . . , is known as an infinite sequence. If m1 + m2 + m3 + m4 + m5 + m6 + . . . . . . + mn = Sn, then Sn is termed as the sum to n elements of the series. 
General Form: \([p_{n}]_{n=1}^{\infty }\). General Form: \(S_{n}=\sum_{r=1}^{n}m_{r}\). 
The order of a sequence matters. Hence, a sequence 5, 6, 7 is different from 7, 6, 5. However, in case of series 5 + 6 + 7 is same as 7 + 6 + 5.

Frequently Asked Questions on Sequence and Series

What is meant by arithmetic sequence?

In maths, arithmetic sequence, also known as the arithmetic progression where the difference between the two consecutive terms in the sequence should be a constant.

Write down the next three terms in the given sequence 1, 4, 7, ….

The next three terms in the sequence are 10, 13, 16. In this sequence, The difference between 1 and 4 is 3, and 4 and 7 is 3. So, the common difference of this sequence is 3. Therefore, 7+3 = 10, 10+3 = 13, 13+3 = 16.

What are the different types of series in maths?

The different types of series in maths are arithmetic series, harmonic series and geometric series, P-series, exponential series and so on.

Define the finite sequence.

If the number of terms in the sequence is finite or fixed length, then it is called a finite sequence.

Define series with an example

The series is defined as the addition/sum of terms in a sequence. For example, 2, 4, 6, 8 is a sequence, then the series is written as 2+4+6+8.

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