Arithmetic Sequences & Series (Edexcel IGCSE Further Pure Maths)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
Arithmetic Sequences
What is an arithmetic sequence?
In an arithmetic sequence, the difference between consecutive terms in the sequence is constant
This means a common difference is added to each term to get the next term
The first term of the sequence is denoted by
The common difference is denoted by
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, , is 1
The common difference, , is 3
An arithmetic sequence can be
increasing (positive common difference), or
decreasing (negative common difference)
Terms in an arithmetic sequence can be referred to
by the letter with
a subscript corresponding to its place in the sequence
e.g. is the first term, is the ninth term, is the th term, etc.
Arithmetic Series
What is an arithmetic series?
When the terms of an arithmetic sequence are added together, that is known as an arithmetic series
The terms (1st term, 2nd term, 3rd term, etc.) are exactly the same in the sequence and series
But with series we're most interested in what happens when the terms are added together
How do I find a term in an arithmetic series?
The nth term formula for an arithmetic sequence is
Where is the first term, and is the common difference
This is not given on the exam formula sheet, so make sure you know it
The formula allows you to find any term in the arithmetic series
Enter the values of , and and calculate the value of
Sometimes you will be given a term () and asked to find or
Substitute the information you have into the formula and solve the equation
Sometimes you will be given two terms and asked to find both and
Substitute the information into the formula and set up a pair of simultaneous equations
Then solve the simultaneous equations
How do I find the sum of an arithmetic series?
An arithmetic series is the sum of the terms in an arithmetic sequence
For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …
Use the following formula to find the sum of the first terms of an arithmetic series:
is the first term
is the common difference
The formula is given on the exam formula sheet
So you don't need to remember it
But you do need to know how to use it!
A question will often give you the sum of a certain number of terms and ask you to find the value of or
Substitute the information you have into the formula and solve the equation
Examiner Tips and Tricks
The formula for the sum of an arithmetic series is on the exam formula sheet
But the nth term formula is not on the formula sheet
Simultaneous equations are often needed within arithmetic series questions
Make sure you are confident solving them!
Worked Example
The fourth term of an arithmetic series is 10 and the ninth term is 25. Find the first term and the common difference of the series.
Put the information for the fourth and ninth terms into the nth term formula
For
For
That gives us two simultaneous equations in and
Subtract the equation from the equation to eliminate
Substitute that value into the first equation to find
That is all the information we need to answer the question
Worked Example
The sum of the first 10 terms of an arithmetic series is 630.
The first term is 18.
a) Find the common difference, , of the series.
Use the arithmetic series formula
Here , and
Substitute in and solve for
The sum of the first 10 terms of another arithmetic series is also 630.
The common difference is 11.
b) Find the first term, , of the series.
Here , and
Last updated:
You've read 0 of your 5 free revision notes this week
Sign up now. It’s free!
Did this page help you?