Linear number sequences

In a nutshell

A linear sequence of numbers consists of a series of numbers that follow a certain pattern. Each number is related to the next in some way and this relation is described by a rule.

Rules for number sequences

There are four rules that you need to know involving adding, taking away, multiplying and dividing.

Rule 1: Add or subtract by the same number

This is where the same number is added or subtracted to give the next term in the sequence.

Example 1

$\begin{array}{ccccccc}&&&+5&&+5&&+5&&+5&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&2&&7&&13&&18&&23& \\&&|&&|&&|&&|&&|&&\\ \hline\end{array}$

In this case, the rule is: "Add five to the previous number".

Rule 2: Multiply or divide by the same number

This is where the same number is multiplied by or divided by to give the next term in the sequence.

Example 2

$\begin{array}{ccccccc}&&&\div2&&\div2&&\div2&&\div2&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&400&&200&&100&&50&&25& \\&&|&&|&&|&&|&&|&&\\ \hline\end{array}$

In this case, the rule is: "Divide the previous number by two".

Rule 3: Add or subtract by a changing number

This is where the number being added or subtracted changes by the same amount each time it is added/subtracted to a term.

Example 3

$\begin{array}{ccccccc}&&&&+1&&+1&&+1&&&\\&&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&&\\&&&+1&&+2&&+3&&+4&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&2&&3&&5&&8&&12& \\&&|&&|&&|&&|&&|&&\\ \hline\end{array}$

In this case, the rule is: "Add one extra to the previous number".

Rule 4: Add the previous two terms

This is where the previous two terms are added to give the next.

Example 4

$\begin{array}{ccccccc}&&&0+1&&1+1&&2+1&&3+2&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&1&&1&&2&&3&&5& \\&&|&&|&&|&&|&&|&&\\ \hline\end{array}$

In this case, the rule is "Add the previous two terms"

Finding missing numbers in a sequence

Using the previous/next terms

You can use the rule you have found to find further numbers in the sequence.

Example 5

Find the missing term in the sequence.

Find the rule.

"Add three to the previous number"

Work out the missing term.

$14+3=\underline{17}$

Using a general rule

A quick way to find any number in the sequence is to work out a general rule. Usually the letter n is used in place of the term you want to find.

PROCEDURE

1.	Find the rule for the sequence.
2.	Use the following: In a sequence where you add the same number use that number multiplied by $n$ . In a sequence where you subtract the same number use $-$ that number multiplied by $n$ .
3.	Look at the first term and $n=1$ to decide what should be taken away or added to the first part of your general rule.
4.	Use your general rule to find any term in the sequence.

Example 6

Find the 30th term in the sequence:

$\begin{array}{ccccccc}&&&+3&&+3&&+3&&+...&\\&&&\curvearrowright&&\curvearrowright&&\curvearrowright&&\curvearrowright&\\&&2&&5&&8&&11&&...& \\&&|&&|&&|&&|&&|&&\\ \hline\end{array}$ 

Find the rule.

"Add three to the previous term"

Multiply n by three.

$3n$

Substitute in one in place of n to find out the change needed.

$3\times1=3\\3-1=2$ 

Take one away from $3n$ to give the general rule.

$3n - 1$

Replace n with $30$ to work out the thirtieth term.

$(3\times30)-1 = \underline{89}$