129 LearnersLast updated on July 4th, 2025
Sequence and Series
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.
What are Sequences and Series?
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 + …
Difference Between Sequence and Series
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 + … |
What are the Types of Sequences and Series?
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:
- Arithmetic Sequence and Series
- Geometric Sequence and Series
- Harmonic Sequence and Series
Arithmetic Sequence and Series
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 Sequence and Series
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.
Harmonic Sequence and Series
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.
Real-world applications of Sequence And Series
Sequences and series are used for various real-life situations. In this section, let’s learn a few applications of sequences and series.
- To calculate interest, loan amortization, and investment growth, we use a geometric sequence.
- To understand the population dynamics, we use sequences and series as they can model the population growth, the spread of a disease, and so on.
- To understand the data structure and algorithms in computer science.
- In biology, the Fibonacci sequence is used to analyze natural phenomena.
Common Mistakes and How to Avoid Them in 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.
Mistake 1
Confusion Between Arithmetic and Geometric Sequences
Confusion with the common difference in arithmetic sequences and ratio in geometric sequences is common among students. So, to avoid this confusion, students should know the patterns, common differences, and ratios. Common difference (d = a_n - a_(n-1)) is used in arithmetic sequences, while the common ratio (r = a_n / a_(n-1)) is used in geometric sequences.
Mistake 2
Using the Wrong Formula for Sum or nth Term
Students are confused in finding the nth term of geometric and arithmetic sequence.
Nth term of arithmetic sequence is calculated by an = a + (n - 1)d
where, ‘an’ is the nth term to be calculated, ‘a’ denotes the first term of the sequence
‘n’ denotes the position of the term that needs to be calculated, and ‘d’ denotes the common difference between the two consecutive terms. Similarly, for Geometric sequence, nth term can be calculated as an = a1 * r(n-1)where, ‘an’ denotes nth term, ‘a1’ denotes the first term, ‘r’ denotes the common ratio, and ‘n’ denotes the term position.
Mistake 3
Error in Identifying the First Term
For calculating the first time, it is advisable to double-check the first term. Any erroneous calculations lead to miscalculation of the first term. For example, mistaking 3 as the first term in 5, 8, 11, … (where a_1 = 5) leads to incorrect calculations.
Mistake 4
Confusing Common Difference with Ratio
Sometimes, students tend to confuse common differences with ratios, which can lead to wrong calculations. So, always remember that the common difference is the difference between two terms in an arithmetic sequence; it is calculated as d = an - an-1. The common ratio is the quotient of successive terms: r = an/a(n-1).
Mistake 5
Miscalculating Negative Ratios in Geometric Sequences
When the common ratio is negative, students may incorrectly apply exponents to negative numbers or ignore sign changes. Make sure to track whether each term is positive or negative. For example, in the sequence 2, -6, 18, -54, … (r = -3), each term alternates signs: an = 2 (-3)(n-1).
Solved Examples of Sequences and Series
Problem 1
Find the 15th term of the arithmetic sequence: 3, 7, 11, 15, …
The 15th term is 59
Explanation
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
Problem 2
Find the missing term in the sequence 4, __, 16, 22.
The missing term is 10
Explanation
The common difference = 22 -16 = 6
So, the next term after 4 is 4 + 6 = 10
So the sequence is 4, 10, 16, 22
Problem 3
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
Explanation
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
Problem 4
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
Explanation
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
Problem 5
Find the common ratio of the sequence: 2, 10, 50, 250, …
The common ratio here is 5
Explanation
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
FAQs on Sequences and Series
1.What are the 4 types of sequence?
2.What sequence is 2, 4, 6, 8, 10?
3.How do you calculate the sum of an arithmetic series?
4.What is the formula for the sum of an infinite geometric series?
5.Where are sequences and series used in real life?
6.How can children in Thailand use numbers in everyday life to understand Sequence and Series ?
7.What are some fun ways kids in Thailand can practice Sequence and Series with numbers?
8.What role do numbers and Sequence and Series play in helping children in Thailand develop problem-solving skills?
9.How can families in Thailand create number-rich environments to improve Sequence and Series skills?
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Hiralee Lalitkumar Makwana
About the Author
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.
Fun Fact
: She loves to read number jokes and games.
