Arithmetic Sequences (Edexcel IGCSE Maths A (Modular))

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Mark Curtis

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Maths

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 $a + (11 - 1) d = 11$
  - This simplifies to $a + 10 d = 11$
- Forming two different equations in $a$ and $d$ can lead to simultaneous equations

Exam Tip

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, $d$ .

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

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

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

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

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

$- a + 20 d = 43 a + 5 d = 13 15 d = 30$

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

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

$\begin{array}{rcl} 13 & = & a + 5 \times 2 \\ 13 & = & a + 10 \\ 3 & = & a \end{array}$

Write out the final answer

The first term is 3 and the common difference is 2

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