Sequences and Series: Definition, Formulas, and Examples
A sequence is a list of numbers arranged in a pattern, such as 3, 6, 9, 12. A series is what you get when you add the terms of a sequence together, such as 3 + 6 + 9 + 12 = 30. This page explains the concept of sequences and series in simple language, with formulas, worked examples, and exam-style questions for GCSE, IGCSE, and A Level students.
Quick answer: If you are looking at numbers one after another, you are dealing with a sequence. If you are adding those numbers, you are dealing with a series. In exams, the main types you need are arithmetic and geometric.
What is the difference between a sequence and a series?
What is a sequence?
A sequence is an ordered list of numbers that follows a rule. Each number is called a term. For example, 5, 8, 11, 14 is a sequence because each term goes up by 3.
What is a series?
A series is the sum of the terms of a sequence. For example, if the sequence is 5, 8, 11, 14, then the series is \(5 + 8 + 11 + 14 = 38\).
Memory tip: Sequence = see the numbers. Series = sum the numbers.
Key words students should know
| Word | Meaning | Simple example |
|---|---|---|
| Term | A number in a sequence | In 2, 4, 6, 8, the term 6 is the 3rd term |
| nth term | A rule for finding any term | \(a_n = 2n + 1\) |
| Series | The sum of sequence terms | \(2 + 4 + 6 + 8\) |
| Common difference | Amount added each time | In 7, 10, 13, 16, the difference is 3 |
| Common ratio | Amount multiplied each time | In 3, 6, 12, 24, the ratio is 2 |
Common types of sequences and series
Arithmetic sequence
An arithmetic sequence changes by the same difference each time.
Example: 4, 7, 10, 13, 16
Each term increases by 3, so the common difference is \(d = 3\).
Geometric sequence
A geometric sequence changes by the same ratio each time.
Example: 2, 6, 18, 54
Each term is multiplied by 3, so the common ratio is \(r = 3\).
Quadratic sequence
A quadratic sequence has constant second differences.
Example: 1, 4, 9, 16, 25
This follows the rule \(n^2\).
Fibonacci sequence
Each term is the sum of the two previous terms.
Example: 1, 1, 2, 3, 5, 8, 13
How to identify the type quickly
- If you add the same number each time, it is arithmetic.
- If you multiply by the same number each time, it is geometric.
- If first differences are not constant but second differences are constant, it is usually quadratic.
- If each term uses the previous terms, it may be recursive, like Fibonacci.
Key formulas you need to know
These are the main formulas students use in GCSE, IGCSE, and A Level questions. The short explanation under each one tells you when to use it.
Arithmetic sequence formulas
nth term:
\[ a_n = a_1 + (n - 1)d \]
Use this when the sequence goes up or down by the same amount each time.
Sum of the first n terms:
\[ S_n = \frac{n}{2}\big(2a_1 + (n-1)d\big) \]
or
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
Use this when you need the total of the first \(n\) terms.
Geometric sequence formulas
nth term:
\[ a_n = a_1 r^{\,n-1} \]
Use this when the sequence is multiplied by the same value each time.
Sum of the first n terms:
\[ S_n = a_1 \cdot \frac{1-r^n}{1-r} \quad (r \ne 1) \]
Use this for a finite geometric series.
Infinite geometric series sum:
\[ S_\infty = \frac{a_1}{1-r} \quad \text{when } |r| < 1 \]
Use this only when the common ratio is between -1 and 1.
Sigma notation
\[ \sum_{k=1}^{n} a_k \]
This means “add the terms from the 1st term to the nth term.”
Fully worked examples
Worked Example 1: Find the nth term of an arithmetic sequence
Question: Find the 20th term of the sequence 6, 10, 14, 18, ...
Step 1: Identify the first term and common difference.
\(a_1 = 6\), \(d = 4\)
Step 2: Use the arithmetic nth-term formula.
\[ a_n = a_1 + (n-1)d \]
Step 3: Substitute \(n = 20\).
\[ a_{20} = 6 + (20-1)\cdot 4 \]
\[ a_{20} = 6 + 76 = 82 \]
Answer: The 20th term is 82.
Worked Example 2: Find the sum of an arithmetic series
Question: Find the sum of the first 15 terms of the sequence 3, 7, 11, 15, ...
Step 1: Identify the values.
\(a_1 = 3\), \(d = 4\), \(n = 15\)
Step 2: Find the 15th term.
\[ a_{15} = 3 + (15-1)\cdot 4 = 3 + 56 = 59 \]
Step 3: Use the arithmetic sum formula.
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
\[ S_{15} = \frac{15}{2}(3 + 59) \]
\[ S_{15} = \frac{15}{2}\cdot 62 = 15 \cdot 31 = 465 \]
Answer: The sum of the first 15 terms is 465.
Worked Example 3: Find the sum of a geometric series
Question: Find the sum of the first 6 terms of the sequence 5, 10, 20, 40, ...
Step 1: Identify the first term and common ratio.
\(a_1 = 5\), \(r = 2\), \(n = 6\)
Step 2: Use the geometric sum formula.
\[ S_n = a_1 \cdot \frac{1-r^n}{1-r} \]
Step 3: Substitute the values.
\[ S_6 = 5 \cdot \frac{1-2^6}{1-2} \]
\[ S_6 = 5 \cdot \frac{1-64}{-1} \]
\[ S_6 = 5 \cdot 63 = 315 \]
Answer: The sum of the first 6 terms is 315.
Exam-style questions with brief answers
These short questions are designed to feel like GCSE, IGCSE, or A Level revision tasks.
1. Find the next term in the sequence 12, 17, 22, 27, ...
2. Is the sequence 4, 8, 12, 16 arithmetic or geometric?
3. Find the 12th term of the arithmetic sequence 5, 9, 13, 17, ...
4. Find the common ratio of the sequence 3, 12, 48, 192, ...
5. Find the 7th term of the geometric sequence 2, 6, 18, 54, ...
6. Write the nth term of the arithmetic sequence 8, 13, 18, 23, ...
7. Find the sum of the first 10 terms of 1, 4, 7, 10, ...
8. Does the infinite geometric series \(9 + 3 + 1 + \frac{1}{3} + \cdots\) converge?
9. What is the sum to infinity of \(9 + 3 + 1 + \frac{1}{3} + \cdots\)?
10. Which type of sequence is 1, 4, 9, 16, 25, ... ?
Common mistakes students make
Mixing up sequence and series: A sequence lists terms; a series adds them.
Using the wrong formula: Arithmetic uses a common difference. Geometric uses a common ratio.
Forgetting the \((n-1)\): In nth-term formulas, students often use \(n\) instead of \((n-1)\).
Using a sum to infinity when the ratio is too large: You can only use \(S_\infty = \frac{a_1}{1-r}\) if \(|r| < 1\).
Not checking what the question asks: Some questions want the nth term, others want the sum.
Interactive sequence and series calculator
Arithmetic sequence calculator
Enter the first term, common difference, and term number.
Geometric sequence calculator
Enter the first term, common ratio, and term number.
Frequently asked questions
What is the difference between a sequence and a series?
A sequence is a list of numbers in order. A series is the sum of those numbers. For example, 2, 4, 6, 8 is a sequence, while \(2 + 4 + 6 + 8 = 20\) is the series.
How do I know if a sequence is arithmetic?
Check the difference between consecutive terms. If the difference is always the same, the sequence is arithmetic.
How do I know if a sequence is geometric?
Check the ratio between consecutive terms. If each term is multiplied by the same number to get the next term, it is geometric.
What is the nth term and why is it useful?
The nth term is a formula that lets you find any term in a sequence without writing every previous term. It saves time and is very common in exam questions.
Can an infinite series have a finite sum?
Yes. An infinite geometric series can have a finite sum if the common ratio satisfies \(|r| < 1\). In that case, the terms get smaller and approach zero.
