Last updated on July 4th, 2025
Sequences and series are two closely related mathematical concepts about patterns of numbers. There are different types of sequences and series. Let's learn more about them in this article.
A sequence of a number is always arranged in a specific pattern. The numbers in the sequence are known as terms. The terms in a sequence may repeat depending on the pattern. A series is the sum of a sequence’s terms, denoted with ‘+’ signs between terms. For example, 2, 4, 6, 8, 10, 12, …, is an arithmetic sequence with a common difference of 2. The corresponding series is 2 + 4 + 6 + 8 + 10 + 12 + …
Although sequences and series are closely connected, they do have their differences. Let's check them out in the table given below:
Sequence | Series |
A sequence is a continuous list of numbers with a specific pattern |
Series is the sum of the terms found in the sequence |
In sequence, the order of the terms is important |
In a series, the sum’s value is independent of the order of addition, but the sequence’s pattern determines the terms. |
The sequence follows a general form: {an}n = 1 |
The series follow the general form: Sn = r = 1nar |
For example: 1, 3, 5, 7, 9, … |
For example: 1 + 3 + 5 + 7 + 9 + … |
There are different types of sequences and series. In this section, we will learn more about the types of sequences and series. Sequences and series are classified into types:
A sequence of numbers is said to be an arithmetic sequence when each term increases or decreases by a constant difference. The difference between the terms is known as a common difference; it is denoted as d. The arithmetic sequence follows the format of a, a + d, a + 2d, a + 3d, … The formula to find the nth term of the arithmetic sequence is an = a1 + (n -1) d. For example, 5, 10, 15, 20, 25, … here the common difference is 5.
The arithmetic series is the series formed by adding the arithmetic sequence. The arithmetic series follows the format of a + (a + d) + (a + 2d) + (a + 3d) + …For example, the sequence 5, 10, 15, 20, 25, … has a series 5 + 10 + 15 + 20 + 25 + …
The formula to find the sum of arithmetic series is Sn = n/2 (2a + (n -1)d)
Geometric sequences have a common ratio between the successive terms. The geometric sequence follows the pattern of a, ar, ar2, …, ar(n - 1), where r is the common ratio and “a” is the first term. For example, 2, 4, 8, 16, … The series formed using the geometric sequence is the geometric series. The geometric series can be represented by a + ar + ar2 + … + ar(n - 1). For example, 2 + 4 + 8 + 16 + … The formula used to find the nth term of the geometric sequence is an = arn -1, and the formula for the sum of geometric series is Sn = a1-rn/1-r for a finite series. We use the formula Sn = a/(1-r) for infinite series if r<1.
A harmonic sequence is the reciprocals of an arithmetic sequence’s terms (e.g., 1/5, 1/10, 1/15, 1/20, …, where 5, 10, 15, 20, … is arithmetic). The nth term of a harmonic sequence is an = 1/a1+(n-1)d, where a1, d are from the arithmetic sequence. The harmonic series sum has no simple closed form.
Sequences and series are used for various real-life situations. In this section, let’s learn a few applications of sequences and series.
Like any other mathematical concepts, sequences, and series can also be quite tricky to perfect. But with the right knowledge and practice, we can easily master these concepts. This section covers common mistakes and how to avoid them.
Find the 15th term of the arithmetic sequence: 3, 7, 11, 15, …
The 15th term is 59
The arithmetic sequence follows the formula: an = a + (n - 1)d
Here, a = 3
Each term increases by 4
So, d = 7 - 3 = 4
Therefore, the 15th term: a15 = 3 + (15 - 1) × 4
= 3 + 14 × 4 = 59
Find the missing term in the sequence 4, __, 16, 22.
The missing term is 10
The common difference = 22 -16 = 6
So, the next term after 4 is 4 + 6 = 10
So the sequence is 4, 10, 16, 22
Find the sum of the first 5 terms of the geometric series: 1, 3, 9, 27, …
The sum of the first 5 terms is 121
The sum of a geometric series is Sn = a(rn - 1)/(r - 1)
Here, a = 1
r = 3
n = 5
S5 = 1 × (35 - 1)/(3 - 1)
= 1 × (243 - 1)/(3 - 1)
= 242 / 2 = 121
Find the sum of the first 20 terms of the arithmetic series 1, 3, 5, 7, …
The sum of the first 20 terms is 400
The sum of the first n terms = n/2 (2a + (n - 1) d)
Where a = 1
d = 2
n = 20
S20 = 20/2 (2 x 1) + ((20 - 1) 2)
= 10 × (2 + 38)
= 10 × 40 = 400
Find the common ratio of the sequence: 2, 10, 50, 250, …
The common ratio here is 5
The common ratio is the ratio between the two consecutive terms; here r = 10/2 = 5
Verify: 50/10 = 250/50 = 5.
So, the common ratio is 5
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.