Sequences & Series
Edexcel International A Level Maths: Pure 2
The Big Idea: A sequence is an ordered list of numbers following a rule. A series is the sum of those numbers. We can describe sequences using general term formulas, sigma notation, or recurrence relations — they're all different ways of saying the same thing.
Quick Overview
What You Need to Know
- Sequences are ordered lists with a rule (e.g. 1, 4, 7, 10, ...)
- Series are the sums (e.g. 1 + 4 + 7 + 10)
- Classify sequences: arithmetic, geometric, increasing, decreasing, or periodic
- Sigma notation (Σ) is compact notation for a sum
- Recurrence relations define each term using the previous one
- Arithmetic and geometric sequences have standard formulas; other sequences need creative approaches
1. Language of Sequences & Series
What is a Sequence?
A sequence is an ordered list of numbers where there's a rule that tells you how to find each number. Think of it as a list where every item has its place, and position matters.
For example: 1, 4, 7, 10, 13, ...
Here's the rule: "Start at 1, then add 3 each time." Every term has a specific position:
Position: 1st 2nd 3rd 4th 5th
Term: 1 4 7 10 13
Notation: u₁ u₂ u₃ u₄ u₅
We call the terms u₁, u₂, u₃, ... or uₙ where n tells us the position. This is just notation—it helps us talk about "the 5th term" without writing out the whole sequence.
The General Term (or Nth Term)
This is a formula that works for any position. For our sequence, the general term is uₙ = 3n − 2. Let's check: u₁ = 3(1) − 2 = 1 ✓, u₂ = 3(2) − 2 = 4 ✓, u₃ = 3(3) − 2 = 7 ✓
What is a Series?
A series is what you get when you add up all the terms in a sequence.
Sequence: 1, 4, 7, 10, 13, ...
Series: 1 + 4 + 7 + 10 + 13 + ...
We use Sₙ to mean "the sum of the first n terms." So:
Classifying Sequences
Increasing & Decreasing Sequences
Increasing Sequence: Each term is larger than the one before it. Written as uₙ₊₁ > uₙ.
Example: 1, 4, 7, 10, 13, ... (goes up by 3 each time)
Decreasing Sequence: Each term is smaller than the one before it. Written as uₙ₊₁ < uₙ.
Example: 20, 16, 12, 8, 4, ... (goes down by 4 each time)
But not all sequences are increasing or decreasing! The sequence 1, −1, 1, −1, ... goes up and down, so it's neither.
Periodic Sequences
Periodic Sequence: The terms repeat in a predictable cycle. The order (or period) is how many terms before the pattern repeats.
Examples of Periodic Sequences
Sequence 1: 2, 3, 2, 3, 2, 3, ...
The pattern {2, 3} repeats. Order = 2.
Sequence 2: 1, 2, 3, 1, 2, 3, 1, 2, 3, ...
The pattern {1, 2, 3} repeats. Order = 3.
Sequence 3: 5, 5, 5, 5, ...
The pattern {5} repeats (constant). Order = 1.
Periodic Sequences in Disguise!
Practice: Classify Sequences
2. Sigma Notation
What is Sigma Notation?