Year 8 Interactive Maths - Second Edition

## Number Patterns

A sequence is a pattern of numbers that are formed in accordance with a definite rule.

We can often describe number patterns in more than one way.  To illustrate this, consider the following sequence of numbers {1, 3, 5, 7, 9, …}.

Clearly, the first term of this number pattern is 1; and the terms after the first term are obtained by adding 2 to the previous term.  We can also describe this number pattern as a set of odd numbers.

By trial and error, we find that:

By observation, we notice that we can describe this number pattern by the rule

### Formula and Tables

A table of values can be generated from the rule

as shown below.

## Finding the Algebraic Rule

We use algebra to study rules that describe the behaviour of everyday things.  For example, the behaviour of the height of a ball when it is thrown upward or the amount outstanding for a loan after a number of regular repayments.  By finding a pattern in observed values (i.e. measurements), we are often able to discover a rule that allows us to make accurate predictions.

### Using a Difference Pattern

When we try to discover an algebraic rule for ordered pairs, we can find the difference between two successive values of y.  This allows us to find a rule as illustrated below.

Consider the following table.

We notice that the values of x increase by just one at a time and the difference between the successive values for y is 2.  So, the rule starts off with y = 2x.  Will this give a correct answer from the table?  Let us check.

The answer is no.  From the table, when x = 1 the value of y should be 5.  How do we change our answer from 2 to 5?  We should add 3.

Check the rule to see if it is correct:

#### Example 5

Discover the rule for the following table of values:

##### Solution:

In the given table, the x-values increase by 1 for each ordered pair.

Find the difference between the successive values of y.  That is:

The difference between successive values of y is always 3.  So, the rule is of the form

###### Check:

Check the rule to see if it is correct:

So, our rule is correct.

###### Note:

To establish a rule for a number pattern involving ordered pairs of x and y, we can find the difference between every two successive values of y.  If the difference pattern is the same, then the coefficient of x in the algebraic rule (or formula) is the same as the difference pattern.