Introduction to Sequences
Learn what sequences are, how to identify patterns, and write rules for finding any term.
Definition
- = first term
- = second term
- = the th term (general term)
- (first term)
- (second term)
- The pattern: add 3 to get the next term
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Worked Examples
Find the pattern and the next three terms:
Find the difference between consecutive terms
, , → Common difference = 4
Identify the pattern
Each term is 4 more than the previous term → Add 4 to get next term
Find the next three terms
, , →
Answer: Pattern: Add 4. Next three terms:
Common Mistakes
Confusing the term number with the term value
Why it's wrong: In , students mix up 5 (position) with 17 (value).
Correct: means the value of the term at position . If , then the 5th term has a value of 17.
Using instead of in the formula
Why it's wrong: The formula gives wrong answers because it counts the difference one extra time.
Correct: Use because you add the difference times to reach the th 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.
Interactive Visual
Click on numbers to select them. Adjust the range to explore different values.
Interactive Sandbox
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History
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Practice Problems
18 problemsWhat is the next term in the sequence ?
Why It Matters
- 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
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 months: euros.
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 hours:
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 representing the th term
- 3Arithmetic sequences have a constant difference:
- 4Geometric sequences have a constant ratio:
- 5Always verify your formula by checking it produces known terms
Frequently Asked Questions
Glossary
- Sequence
- An ordered list of numbers following a specific pattern or rule
- Term
- Each individual number in a sequence
- First term ()
- The starting number of a sequence
- Common difference ()
- The constant value added between consecutive terms in an arithmetic sequence
- Common ratio ()
- The constant value multiplied between consecutive terms in a geometric sequence
- General term ()
- A formula that gives the value of any term based on its position
Formula Card
Arithmetic Sequence
Where $a_1$ is the first term and $d$ is the common difference
Geometric Sequence
Where $a_1$ is the first term and $r$ is the common ratio