Sequence or series?
Big idea: A sequence is a list of terms. A series is what you get when you add those terms.
| Type | Meaning | Example |
|---|---|---|
| Sequence | List of terms | 3, 7, 11, 15, ... |
| Series | Sum of terms | 3 + 7 + 11 + 15 + ... |
| Arithmetic series | Add terms from an arithmetic sequence | 3 + 7 + 11 + 15 |
Quick checks
- Commas usually show a sequence.
- Plus signs usually show a series.
- If the sequence is arithmetic, the series is an arithmetic series.
Sum of the first n terms
Big idea: Use this when the question wants a total, not just one term.
Quick reminder: If the question asks for one term, use uₙ = u₁ + (n − 1)d. If it asks for a total, use Sₙ.
What each letter means
- Sₙ = the sum of the first n terms
- n = how many terms you are adding
- u₁ = the first term
- d = the common difference
Worked example 1 — straight substitution
Find the sum of the first 5 terms of 2, 5, 8, 11, …
Step by step
- Write down u₁, d, and n.
- Substitute into Sₙ = (n/2)(2u₁ + (n − 1)d).
- Work the brackets first.
- Multiply out.
Final answer
S₅ = 40
Worked example 2 — slightly larger n
Find the sum of the first 12 terms of 7, 11, 15, 19, …
Step by step
- Identify u₁, d, and n.
- Substitute into the formula.
- Inside the brackets: 2·7 = 14, and 11·4 = 44.
- Multiply out.
Final answer
S₁₂ = 348
Exam Tips:
- Write down u₁, d, and n first.
- Do not confuse the nth term with the sum formula.
- Use brackets carefully.
Practice with real exam questions
Answer exam-style questions and get AI feedback that shows you exactly what examiners want to see in a full-marks response.
A second sum formula
Big idea: In the last section, you used Sₙ = (n/2)(2u₁ + (n − 1)d). Now you will learn a second sum formula.
This formula is useful when you know the first term and the last term.
Which formula should you use?
- If you know u₁, d, and n, use Sₙ = (n/2)(2u₁ + (n − 1)d).
- If you know u₁ and uₙ, use Sₙ = (n/2)(u₁ + uₙ).
First look at the information in the question. Then choose the formula that matches it.
Worked example 1 — when d is given
Find the sum of the first 20 terms of 3, 7, 11, 15, 19, …
Step by step
- Identify u₁, d, and n.
- Use the first formula since d is given.
- Simplify the brackets.
- Multiply.
Final answer
S₂₀ = 820
Worked example 2 — when the last term is given
The 1st term of an arithmetic sequence is 14 and the 25th term is 86. Find the sum of the first 25 terms.
Step by step
- Write down what you know.
- Since u₁ and uₙ are given, use the second formula.
- Add inside the brackets.
- Multiply.
Final answer
S₂₅ = 1250
Exam Tips:
- First check what the question gives you.
- Use the first formula when d is given.
- Use the second formula when the last term is given.