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239 LearnersLast updated on December 3, 2025
Sequence and Series
Sequences and series are mathematical ideas based on number patterns. A sequence is an ordered list of numbers that follow a rule, whereas a series is the sum of the numbers in a sequence. In this article, we will explore different types of sequences and series with easy examples.
What are Sequences and Series?
A sequence of numbers is always arranged in a specific pattern. The numbers in the sequence are known as terms. The terms in a sequence may sometimes 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 + …
Definition of Sequence and Series
A sequence is an ordered list of numbers that follow a specific rule or pattern. Each number in the list is called a term of the sequence.
A series is the sum of the terms of a sequence, written with the ‘+’ sign between them.
Examples of Sequences and Series:
- 2, 5, 8, 11, 14, . . . .
Here, the difference between consecutive terms is 3, and it is a constant. This is an arithmetic sequence.
And the corresponding arithmetic series is:
2 + 5 + 8 + 11 + 14 + . . . . .
- 3, 6, 12, 24, 48,...
Here, each term is multiplied by a fixed number to obtain the following number in the sequence. This is a geometric sequence.
And the corresponding geometric series is:
3 + 6 + 12 + 24 + 48 + . . . . .
- 1, 1/2, 1/3, 1/4, 1/5,......
Here, each term is formed by taking the reciprocals of the natural numbers. This is a harmonic sequence. And the corresponding harmonic series is:
1 + 1/2 + 1/3 + 1/4 + 1/5 +….
Difference Between Sequence and Series
Although sequences and series are closely connected, they 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: \(\{ {a_n} \} _{n = 1} ^{\infty}\) |
The series follow the general form: \(S_n = \sum _ { r = 1} ^{n} a_r\) |
|
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 is arithmetic if 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 \(Sₙ = {n\over 2} × [2a + (n - 1)d]\)
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 = ar(n – 1),
- The formula for the sum of geometric series is \(S_n = a [{1 \space - \space r^n \over { 1\space - \space r}}]\) for a finite series. We use the formula \(S_{\infty} = a [{1 \over { 1\space - \space r}}]\), 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 \(a_n = {1 \over a_1} + (n - 1)d\), where a1, d are from the arithmetic sequence. The harmonic series sum has no simple closed form.
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Sequence and Series Formula
Sequence and series helps us understand number patterns in mathematics. To easily calculate sequences and series, we use the following formulas.
Arithmetic sequence and series formula
| Arithmetic sequence formula | a, a + d, a + 2d, a + 3d, … |
| Arithmetic series formula | a + (a + d) + (a + 2d) + (a + 3d) + … |
| Common difference, d | \(a_n - a_n-1\) |
| Nth term \(a_n\) | \(a + (n-1)d\) |
| Sum of arithmetic series \(S_n\) | \((\frac{n}{2})(2a + (n-1)d)\) |
Where,
a is the first term,
n is the number of terms
d is the common difference.
Geometric sequence and series formula
| Geometric sequence | \(a, ar, ar^2,….,ar^{(n-1)},…\) |
| Geometric series formula | \(a + ar + ar^2 + ...+ ar^{(n-1)}+ …\) |
| nth term | \(ar^{(n-1)}\) |
| Sum of geometric series | For finite series: \(S_n = a\frac{(1−r^n)}{(1−r)}\) for r≠1, \(S_n = a_n\), for r = 1. For infinite series: \(S_\infty = \frac{a}{1 - r} \quad \text{for } |r| < 1 \) |
Where,
a is the first term,
r is the common ratio.
Harmonic sequence and series formula
- Harmonic sequence formula: /a, 1/a + d, 1/ a+ 2d, . . . .
- nth term: \(\frac{1}{a+(n-1)d}\)
Trips and Tricks to Master Sequence and Series
Solving sequence and series problems can be complex but can be easily done using the correct approach. Here are a few tips and tricks to effectively solve problems related to sequence and series:
- If a, b, c are in an arithmetic progression, then 2b = a + c.
- If a, b, c are in GP, then the relation between a, b and c can be given as: b2 = a × c.
- If a, b, c are in HP, then \(b = {2ac \over {a +c}}\).
- Before applying the formula for sum, check the value of r. If |r| < 1, then apply the formula: \(S_{\infty} = a [{1 \over { 1\space - \space r}}]\).
- If 'd' is not known, then use the formula \(Sₙ = {n\over 2} × [a + l]\) to calculate the sum in an AP. Here, 'a' is the first term and 'l' is the last term.
- Parents and teachers can help students learn about sequences and series by using real-life examples such as number patterns in calendars, staircases, and page counts.
- Encourage students to identify and extend patterns before calculating formulas, for a better understanding of the concepts.
- Provide visual aids, such as charts, sticky notes, or pattern blocks, that display sequences and series to help students see number relationships.
- Start with simple sequences and gradually introduce formulas once the pattern is evident for students.
- Encourage students by asking questions like, "What comes next?" Or how is the number changing each time?. This will help them in quick calculation and deep understanding.
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. The common difference \(d = a_n - a_{n -1}\) is used in arithmetic sequences, while the common ratio \(r = {a_n \over a_{n-1}}\) 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 \(a_n = a + (n - 1)d\)
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 a Geometric sequence, the nth term can be calculated as \(a_n = a_1 \times r^{n-1}\) where, ‘an' denotes the 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 a1 = 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 \(d = a_n - a_{n-1}\). The common ratio is the quotient of successive terms: \(r = {a_n \over {a_{n-1}}}\).
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.
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, geometric sequences are used.
- 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.
- In physics, sequence and series are used to describes motion of objects, heat transfer, oscillations and are even used in fluid mechanics.
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: \(a_n = a + (n - 1)d\)
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: \( S_n = {a [{(r^n - 1) \over {r - 1}}]}\)
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 = \({n\over 2} {\times [(2a + (n - 1) d)]}\)
Here, a = 1
d = 2
n = 20
S20 = 20/2 × [(2 × 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?
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.
