Grade 12 · Free math lesson
Sequences & Series
After this lesson, you will be able to identify patterns in number lists and find their sums.
Practice Sequences & Series freeSkills in this unit
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- Arithmetic sequences
- Geometric sequences
- Arithmetic series
- Finite geometric series
- Infinite geometric series
- Sigma notation
- Recursive rules and patterns
- Sequences as functions
- The binomial theorem
What Is a Sequence?
A sequence is just a list of numbers that follow a pattern. Each number in the list is called a term.
Example: 2, 4, 6, 8, 10 — each term goes up by 2. The 5th term is 10.
Remember: Think of a sequence like steps on a staircase — each step follows a rule.
Arithmetic vs Geometric
An arithmetic sequence adds the same number each time. A geometric sequence multiplies by the same number each time.
Example: Arithmetic: 5, 10, 15, 20 (add 5 each time). Geometric: 3, 6, 12, 24 (multiply by 2 each time).
Remember: Arithmetic = Add. Geometric = Grow (multiply). A comes before G — just like Add before Grow!
What Is a Series?
A series is what you get when you ADD up the terms of a sequence. Instead of listing terms, you find their total.
Example: Sequence: 1, 2, 3, 4, 5. Series (sum): 1 + 2 + 3 + 4 + 5 = 15.
Remember: Sequence = LIST. Series = SUM. S for Series, S for Sum!
Worked example
Find the 10th term and the sum of the first 10 terms of: 3, 7, 11, 15...
- 1Step 1 — Find the rule: Each term adds 4, so this is arithmetic with first term a = 3 and common difference d = 4.
- 2Step 2 — Find the 10th term: Use the formula T(n) = a + (n−1)d. T(10) = 3 + (10−1)×4 = 3 + 36 = 39.
- 3Step 3 — Find the sum of 10 terms: Use S(n) = n/2 × (first term + last term). S(10) = 10/2 × (3 + 39) = 5 × 42 = 210.
Answer: The 10th term is 39 and the sum of the first 10 terms is 210.
The #1 mistake to avoid
Students forget to subtract 1 in the formula T(n) = a + (n−1)d and write a + nd instead — always use (n−1), not n!
Key takeaways
- A sequence is an ordered list of numbers that follow a rule.
- Arithmetic sequences add a fixed number; geometric sequences multiply by a fixed number.
- A series is the sum of the terms in a sequence — use the sum formula to find it fast.
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