Arithmetic Sequences (Edexcel IGCSE Maths)

Spec point — 3.1C

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

  • $a$ 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 $u_{3} = 10 + ( 3 - 1 ) \times 4 = 18$

    • 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 $u_{n} = 10 + ( n - 1 ) \times 4$

    • This simplifies to $u_{n} = 10 + 4 ( n - 1 ) = 10 + 4 n - 4 = 6 + 4 n$

• 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 $a + ( 5 - 1 ) d = 11$
    • This simplifies to $a+4d=11$

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

Exam Hint You are not given the formula for the $n$^th term in the formula booklet.

Worked Example The 6^th and 21^st terms in an arithmetic sequence are 13 and 43, respectively.

Find the first term, $a$, and the common difference, $d$.

Answer:

The formula for an arithmetic sequence is $u_{n} = a + ( n - 1 ) d$ If the 6^th term is 13, then $u_{6}=13$ Substitute $n=6$ and $u_{6}=13$ into the formula

$\begin{gathered} u_{6} & = a + ( 6 - 1 ) d \\ 13 = a + 5 d \end{gathered}$

Similarly, if the 21^st term is 43, then $u_{21}=43$ Substitute $n=21$ and $u_{21}=43$ into the formula

$\begin{gathered} u_{21} & = a + ( 21 - 1 ) d \\ 43 = a + 20 d \end{gathered}$

These give two different equations in $a$ and $d$ Solve the equations simultaneously (for example, eliminate $a$)

- charalign center stackalign right end | a + 20 d = 43 end | | a + 5 d = 13 end | horizontal line | 15 d = 30 end | end

$d=\frac{30}{15}=2$

Substitute $d=2$ into the first equation and solve for $a$

$\begin{gathered} 13 = a + 5 \times 2 \\ 13 = a + 10 \\ 3 = a \end{gathered}$

Write out the final answer

Final answer: The first term is 3 and the common difference is 2