Introduction to Sequences

Learn what sequences are, how to identify patterns, and write rules for finding any term.

Advanced25 minLesson
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Definition

A sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in the sequence is called a term.
We write sequences using notation:
  • a1a_1 = first term
  • a2a_2 = second term
  • ana_n = the nnth term (general term)
Example sequence: 2,5,8,11,14,…2, 5, 8, 11, 14, \ldots
Here:
  • a1=2a_1 = 2 (first term)
  • a2=5a_2 = 5 (second term)
  • The pattern: add 3 to get the next term
an=2+3(n−1)=3n−1a_n = 2 + 3(n-1) = 3n - 1

Try it now

What is the next term in the sequence 2,5,8,11,…2, 5, 8, 11, \ldots?

Worked Examples

Find the pattern and the next three terms: 3,7,11,15,…3, 7, 11, 15, \ldots
1

Find the difference between consecutive terms

7−3=47 - 3 = 4, 11−7=411 - 7 = 4, 15−11=415 - 11 = 4 → Common difference = 4
2

Identify the pattern

Each term is 4 more than the previous term → Add 4 to get next term
3

Find the next three terms

15+4=1915 + 4 = 19, 19+4=2319 + 4 = 23, 23+4=2723 + 4 = 27 → 19,23,2719, 23, 27

Common Mistakes

Confusing the term number with the term value

Why it's wrong: In a5=17a_5 = 17, students mix up 5 (position) with 17 (value).

Correct: ana_n means the value of the term at position nn. If a5=17a_5 = 17, then the 5th term has a value of 17.

Using nn instead of (n−1)(n-1) in the formula

Why it's wrong: The formula an=a1+nda_n = a_1 + nd gives wrong answers because it counts the difference one extra time.

Correct: Use an=a1+(n−1)da_n = a_1 + (n-1)d because you add the difference (n−1)(n-1) times to reach the nnth term.

Assuming all sequences are arithmetic

Why it's wrong: Not all sequences have a constant difference. Some multiply (geometric) or follow other patterns.

Correct: Always check both differences AND ratios between terms to identify the type of sequence.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Sequences Explained | Finding Terms, nth Term, and Partial Sums

by Understand The Math

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Interactive Visual

Sequence Explorer

2,
5,
8,
11,
14,
17,
20,
23
, ...

Formula

a_n = 2 + 3(n - 1) = 3n - 1
Sum Formula: S_8 = 8/2 × (2×2 + 7×3) = 100
12345678
a_1

2

a_5

14

a_10

29

a_20

59

Sequence Explorer

2,
6,
18,
54,
162,
486,
1458
, ...

Formula

a_n = 2 × 3^(n-1)
1234567
a_1

2

a_5

162

a_10

39,366

a_20

2,324,522,934

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is the next term in the sequence 2,5,8,11,…2, 5, 8, 11, \ldots?

Why It Matters

Sequences are everywhere in mathematics and real life:
  • Finance: Compound interest grows in a geometric sequence
  • Nature: The Fibonacci sequence appears in flower petals, pinecones, and shells
  • Computer Science: Algorithms often process data in sequences
  • Population Growth: Populations grow following sequence patterns
  • Music: Time signatures and rhythm patterns form sequences
Understanding sequences is essential for calculus, statistics, and many areas of advanced mathematics!

Real World Applications

Saving Money

If you save a fixed amount each month, your total savings forms an arithmetic sequence.

Example:

Save 100 euros per month: 100, 200, 300, 400, ... After nn months: an=100na_n = 100n euros.

1Try It Yourself

You start with 50 euros and save 75 euros each month.

How much will you have after 12 months?

Step 1: Write the mathematical expression

Use the formula: First amount + (months - 1) times monthly savings

Bacterial Growth

Bacteria often double every hour, creating a geometric sequence.

Example:

Starting with 100 bacteria that double hourly: 100, 200, 400, 800, ... After nn hours: an=100×2n−1a_n = 100 \times 2^{n-1}

2Try It Yourself

A colony starts with 50 bacteria and triples every hour.

How many bacteria after 5 hours?

Step 1: Write the mathematical expression

Initial amount times ratio raised to (hours - 1)

Key Takeaways

  • 1A sequence is an ordered list of numbers following a pattern
  • 2Each number is called a term, with ana_n representing the nnth term
  • 3Arithmetic sequences have a constant difference: an=a1+(n−1)da_n = a_1 + (n-1)d
  • 4Geometric sequences have a constant ratio: an=a1×rn−1a_n = a_1 \times r^{n-1}
  • 5Always verify your formula by checking it produces known terms

Frequently Asked Questions

A sequence is an ordered list of numbers (e.g., 2, 4, 6, 8). A series is the SUM of a sequence's terms (e.g., 2 + 4 + 6 + 8 = 20).
A sequence is an ordered list of numbers (e.g., 2, 4, 6, 8). A series is the SUM of a sequence's terms (e.g., 2 + 4 + 6 + 8 = 20).
Yes! For example, 5,2,−1,−4,…5, 2, -1, -4, \ldots is an arithmetic sequence with common difference −3-3.
Because the first term doesn't need any additions or multiplications. To get to a2a_2, we add once; to a3a_3, twice; to ana_n, we add (n−1)(n-1) times.

Glossary

Sequence
An ordered list of numbers following a specific pattern or rule
Term
Each individual number in a sequence
First term (a1a_1)
The starting number of a sequence
Common difference (dd)
The constant value added between consecutive terms in an arithmetic sequence
Common ratio (rr)
The constant value multiplied between consecutive terms in a geometric sequence
General term (ana_n)
A formula that gives the value of any term based on its position nn

Formula Card

Arithmetic Sequence

an=a1+(n−1)da_n = a_1 + (n-1)d

Where $a_1$ is the first term and $d$ is the common difference

Geometric Sequence

an=a1×rn−1a_n = a_1 \times r^{n-1}

Where $a_1$ is the first term and $r$ is the common ratio

Today's challengeOne problem, three tries

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Geometric Sequences

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Fibonacci Sequence

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