Arithmetic Sequences & Series
Edexcel International A Level Maths — Pure 2
The big idea: sequences where you always add the same number, and formulas that let you find their sums instantly.
What You'll Learn
📌 Arithmetic Sequences
Recognise sequences where the difference between consecutive terms is constant (the common difference). Use the formula u_n = a + (n−1)d to find any term instantly.
➕ Arithmetic Series
Learn two formulas for summing arithmetic sequences. Understand why they work (the proof is exam-testable).
🔍 Finding a and d
When you know two terms, set up simultaneous equations to find the first term (a) and common difference (d). These unlock everything else.
📋 Real Applications
Arithmetic sequences model linear growth: saving money monthly, building patterns, temperature changes. The sum formula solves "how much total?" instantly.
Part 1: Arithmetic Sequences
What is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always the same. That fixed difference is called the common difference, written as d.
Think of it like climbing stairs at a constant pace. If each stair is 0.3 metres high, then after 1 stair you're 0.3 m up, after 2 stairs you're 0.6 m up, after 3 stairs you're 0.9 m up. The difference between each height is always 0.3 m. That's d.
Examples of arithmetic sequences:
• 2, 5, 8, 11, 14, ... — difference is +3 (d = 3)
• 2, 0.5, −1, −2.5, −4, ... — difference is −1.5 (d = −1.5)
• 7, 7, 7, 7, ... — difference is 0 (d = 0). Yes, a constant sequence is arithmetic!
The nth Term Formula
Once you know the first term (a) and the common difference (d), you can find any term without listing all the ones before it. That's the power of a formula.
Why does it work? Start with a. The 2nd term is a + d. The 3rd term is a + d + d = a + 2d. The 4th term is a + 3d. So the nth term is a + (n − 1) × d. You multiply by (n − 1) because you add d exactly (n − 1) times.
Worked Example 1: Finding a Term
Example
The sequence 2, 5, 8, 11, ... is arithmetic. Find the 20th term.
Identify a and d
First term: a = 2
Common difference: d = 5 − 2 = 3
Apply the formula
un = a + (n − 1)d
u20 = 2 + (20 − 1) × 3
u20 = 2 + 19 × 3
u20 = 2 + 57 = 59
Finding a and d from Two Known Terms
Part 2: Arithmetic Series
What is an Arithmetic Series?
arithmetic series
sum of the first n terms
Sn
Sequence vs. Series:
Sequence:
Series:
The Sum Formulas
When to use which?