Arithmetic Sequences & Series (Edexcel IGCSE Further Pure Maths): Revision Note

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 $a$

• The common difference is denoted by $d$ 

  • 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, $\mathit{a}$, is 1

    • The common difference, $\mathit{d}$, 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 $u$ with

  • a subscript corresponding to its place in the sequence

    • e.g.  $u_1=a$ is the first term, $u_9$ is the ninth term, $u_n$ is the $n$^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 n^th term formula for an arithmetic sequence is

$u_n=a+(n-1)d$

• Where $a$ is the first term, and $d$ 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 $a$, $r$ and $n$ and calculate the value of $u_n$

• Sometimes you will be given a term (${\mathit{u}}_{\mathbf{n}}$) and asked to find $\mathit{a}$ or $\mathit{d}$

  • Substitute the information you have into the formula and solve the equation

• Sometimes you will be given two terms and asked to find both $\mathit{a}$ and $\mathit{d}$

  • 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 $n$ terms of an arithmetic series:

$S_n=\frac{n}{2}\left[2a+(n-1)d\right]$   

• $a$ is the first term

• $d$ 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 $\mathit{a}$ or $\mathit{d}$

  • Substitute the information you have into the formula and solve the equation