Arithmetic Progressions (Cambridge O Level Additional Maths)

Revision Note

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Last updated

12 September 2023

In an arithmetic progression (also called arithmetic sequence), the difference between consecutive terms in the sequence is constant
This constant difference is known as the common difference, d, of the sequence
For example, 1, 4, 7, 10, … is an arithmetic sequence with the rule ‘start at one and add three to each number’
- The first term, a, is 1
- The common difference, d, is 3
An arithmetic progression can be increasing (positive common difference) or decreasing (negative common difference)
Each term of an arithmetic progression is referred to by the letter u with a subscript determining its place in the sequence

Examples of arithmetic progressions

$u_{n} = a + (n - 1) d$

- Where $a$ is the first term, and $d$ is the common difference
- This is given on the list of formulas page of the exam, you do not need to know how to derive it
Sometimes you will be given a term and asked to find the first term or the common difference
- Substitute the information into the formula and solve the equation
Sometimes you will be given two terms and asked to find both the first term and the common difference
- Substitute the information into the formula and set up two simultaneous equations
- Solve the simultaneous equations

Simultaneous equations are often needed within arithmetic progression questions
- Make sure you are confident solving them

Arithm Seq Example, A Level & AS Level Pure Maths Revision Notes

An arithmetic series is the sum of the terms in an arithmetic progression
- It is often referred to as the sum of an arithmetic progression
- For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …

Use the following formulae to find the sum of the first n terms of the arithmetic series:

$S_{n} = \frac{1}{2} n \{a + l\} = \frac{1}{2} n \{2 a + (n - 1) d\}$

- - $a$ is the first term
  - $l$ is the last term
  - $d$ is the common difference
  - $n$ is the number of terms in the series
- Both formulae are given on the formula page, you do not need to know how to derive them
You can use whichever formula is more convenient for a given question
- If you know the first term and common difference use the second version
- If you know the first and last term then the first version is easier to use
A question will often give you the sum of a certain number of terms and ask you to find the value of the first term or the common difference
- Substitute the information into the formula and solve the equation

The arithmetic series formulae are given on page 2 of the exam paper – you don't need to memorise them
- Practise finding the formulae so that you can quickly locate them in the exam