Represent patterns using tables

A table represents a pattern by pairing each input, such as a term number, with its corresponding output, making the relationship easier to trace, extend, and compare. The learner interprets changes in the entries to identify additive or multiplicative rules and describe straightforward relationships with words or algebraic expressions, understanding that one consistent rule must account for every pair. Formal function notation and complex nonlinear or recursive analysis are beyond this scope.

Detailed Explanation: Represent patterns using tables

A table shows how each input is connected to its output. The input might be a term number, and the output might be the value of that term.

Example

A pattern begins:

5,\ 8,\ 11,\ 14,\ldots

Make a table showing the first four term numbers and their values.

Term number (input)	Value (output)
$1$	$5$
$2$	$8$
$3$	$11$
$4$	$14$

Now compare consecutive outputs:

$8-5=3$
$11-8=3$
$14-11=3$

The output increases by $3$ each time, so the pattern follows a consistent additive rule.

To describe the rule, start with $3$ times the term number:

For term $1$ : $3(1)=3$ , and $2$ more gives $5$ .
For term $2$ : $3(2)=6$ , and $2$ more gives $8$ .
For term $3$ : $3(3)=9$ , and $2$ more gives $11$ .

So the rule is

\text{output}=3(\text{term number})+2.

Using $n$ for the term number, this can be written as

3n+2.

To find the fifth term, substitute $n=5$ :

3(5)+2=15+2=17.

The completed table is:

Term number	Value
$1$	$5$
$2$	$8$
$3$	$11$
$4$	$14$
$5$	$17$

Always check that one rule works for every input-output pair in the table.

Learn by doing: Represent patterns using tables

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Input/Output Tables (Basic Operations) - Full Table to Rule

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