Modelling with Sequences & Series

Revise Modelling with Sequences & Series for Pure Mathematics 2 WMA12 (AS Level) — revision notes and instant AI marking.

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Modelling with Sequences & Series

Edexcel International A Level Maths: Pure 2

Real-world situations that change in regular patterns (savings, profits, population growth) can be solved using arithmetic or geometric sequences—you just need to recognize which type fits.

Quick Summary

The Problem: Many exam questions hide sequence and series problems in realistic contexts—savings accounts, business profits, salaries. You need to spot them and solve them.

Arithmetic Models: Use when a fixed amount is added or subtracted each time (e.g., saving £10 extra each month).

Geometric Models: Use when a fixed percentage increase happens or a fixed multiplier is applied (e.g., profit grows by 10% each year).

Your Goal: Recognize the pattern → identify the type → apply the correct formula → solve the problem.

Arithmetic Sequences & Series

What is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the same amount (called the common difference) is added or subtracted between each consecutive term. Think of it like climbing stairs—each step up is the same height.

Analogy: Imagine saving money. You put £100 in your account in January, then every month you add exactly £10 more than the month before. Month 1: £100. Month 2: £110. Month 3: £120. That's an arithmetic sequence—the difference between consecutive amounts is always £10.

The first term is usually written as a, and the common difference as d. So if a = 100 and d = 10, the sequence is: 100, 110, 120, 130, ...

nth Term of an Arithmetic Sequence

u_n = a + (n − 1)d

where u_n is the nth term, a is the first term, d is the common difference, and n is the position of the term you want.

For example, if you want the 10th term in the savings sequence (a = 100, d = 10):
u₁₀ = 100 + (10 − 1) × 10 = 100 + 90 = 190
So in month 10, you'll have saved £190.

Arithmetic Series (Sum)

An arithmetic series is the sum of all the terms in an arithmetic sequence. Instead of just listing the numbers, you add them all up. The formula allows you to find the total quickly without adding each term individually.

Sum of an Arithmetic Series

S_n = n/2 (2a + (n − 1)d)
or equivalently
S_n = n/2 (a + l)

where S_n is the sum of the first n terms, a is the first term, d is the common difference, n is the number of terms, and l is the last term.

Worked Example: Vincent's Savings

Vincent puts £100 into his savings account in January. For the rest of the year, he puts in £10 more than the previous month. How much total will he save by the end of the year (12 months)?

This is an arithmetic series: a = 100, d = 10, n = 12

Using the formula: S₁₂ = 12/2 × (2 × 100 + (12 − 1) × 10) = 6 × (200 + 110) = 6 × 310 = £1,860

Practice Question 1

A health club charges £50 for the first month. Each month after that, the charge increases by £2. How much will a member pay in total over their first 10 months?

Geometric Sequences & Series

What is a Geometric Sequence?

geometric sequence common ratio

a

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Also in the full note

How to Recognize & Choose the Right Model
Common Mistakes & How to Avoid Them
What to Memorise
Concepts Checklist
Exam Tips & Mark Scheme Secrets
Challenge Questions
Geometric Series (Sum)
The Decision Tree

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Modelling with Sequences & Series