Arithmetic Sequences (Edexcel IGCSE Maths B): Revision Note

Arithmetic sequences

What is an arithmetic sequence?

• An arithmetic sequence is a sequence where terms increase by the same amount each time

  • The amount it increases by is called the common difference

    • For example: 3, 5, 7, 9, ...

    • The common difference is 2

• Common differences can be negative

  • These arithmetic sequences decrease by the same amount each time

    • For example: 11, 8, 5, 2, -1, ...

    • The common difference is -3

• Arithmetic sequences are also called linear sequences

What is the formula for the nth term of an arithmetic sequence?

• The formula for the $n$^th term of an arithmetic sequence is

  $u_n=a+\left(n-1\right)d$

  • $a$ is the first term

  • $d$ is the common difference

  • $u_n$ is the $n$^th term

    • e.g. $n=5$ gives the fifth term, $u_5$

How do I use the formula for the nth term of an arithmetic sequence?

• You can substitute in the values of $a$, $d$ and $n$ to find a particular term

  • e.g. $a=10$, $d=4$ and $n=3$ gives $u_3=10+\left(3-1\right)\times 4=18$

    • So 18 is the 3^rd term

• You can substitute in the values of $a$ and $d$ to find an expression for $u_n$

  • e.g. $a=10$ and $d=4$ gives $u_n=10+\left(n-1\right)\times 4$

    • This simplifies to $u_n=10+4\left(n-1\right)=10+4n-4=6+4n$

• You can form equations in terms of $a$ and $d$ if given information about terms

  • e.g. If the 5^th term is 11,

    • $u_5=11$ giving $a+\left(5-1\right)d=11$

    • This simplifies to $a+4d=11$

  • Forming two different equations in $a$ and $d$ can lead to simultaneous equations