Algebra · Patterns and Relations · Lesson 3

Finding the Rule of a Pattern or Mapping and Predicting Terms

Adding the difference again and again works for the next few terms, but what about the 50th term or the 100th? In this lesson you will find a rule that links each term to its position (the nth term), use mapping rules such as $x \to 2 x + 3$ , and work backwards to find a missing input. These skills let you predict any term without listing all the ones before it.

NaCCA B7.2.1.1.3 ⏱ about 50 min

By the end you will be able to

✔Find the rule (nth term) of a pattern that changes by the same amount each time.
✔Use a mapping rule to find outputs for given inputs.
✔Work backwards from an output to find the input of a mapping.
✔Predict far-off terms and decide which term has a given value.

Key words

Term number (n): The position of a term in a pattern: 1st, 2nd, 3rd, … We use the letter n for any position.
nth term: A rule that gives the value of a term from its position, e.g. 3n + 1 gives 4, 7, 10, 13, …
Common difference: The fixed amount added (or subtracted) to get from one term to the next.
Mapping: A rule that sends each input to one output, written like $x \to 2 x + 3$ and shown with arrows.
Inverse operation: The operation that undoes another: subtraction undoes addition, division undoes multiplication.

Curriculum reference

Content standard: B7.2.1.1 Derive the rule for a set of points of a relation, draw a table of values to graph the relation in a number plane and make predictions about subsequent elements of the relation.

Indicator: Identify the relation or rule in a pattern/mapping presented numerically or symbolically and predict subsequent elements.

Starter

How many chairs in row 25?

For the Independence Day durbar at your school, the headteacher wants 25 rows of chairs on the field. Row 1 has 12 chairs, Row 2 has 15 and Row 3 has 18. The prefects need to know how many chairs Row 25 will have so they can arrange them from the back first. Writing out all 25 rows takes too long. A rule for the nth row gives the answer in one line.

1. From the difference to the rule

In the last lessons you extended patterns by adding the difference each time. That works for the next few terms, but it is slow for the 25th or 100th term. We want a rule that uses the term number n directly.

Take the pattern 7, 11, 15, 19, … The common difference is 4. That tells us the rule contains 4n — four times the term number. Now compare the pattern with the 4 times table:

Every term is 3 more than 4n, so the rule is 4n + 3.
Term number (n)	1	2	3	4
4n	4	8	12	16
Term	7	11	15	19
Term − 4n	3	3	3	3

1Find the common difference d (here d = 4).
2Write d × n (here 4n) and work out its values for n = 1, 2, 3, ….
3See what you must add or subtract to get each term (here +3).
4Write the rule: nth term = 4n + 3. Test it: n = 3 gives 4 × 3 + 3 = 15 ✓

▶ Each figure adds a new row of 4 dots, so the rule has 4n; the 3 extra dots stay the same in every figure, giving 4n + 3.

2. Rules for matchstick and dot patterns

The same method works for patterns made with objects. Squares in a row use 4, 7, 10, 13 matchsticks. The difference is 3, so the rule has 3n, and 4 − 3 = 1, so sticks = 3n + 1.

You can also see the rule in the picture: there is 1 starting stick on the left, and each square adds a group of 3 sticks (top, bottom and right side). Pattern n has n groups of 3 plus the 1 starting stick: 3n + 1. Seeing the rule in the picture is a great way to check your working.

3. Mappings and their rules

A mapping sends each number in the domain to a number in the co-domain using a rule. We write it with an arrow: $x \to 2 x + 3$ means x maps to two times x, plus 3.

▶ Follow each arrow: every input is doubled and then 3 is added.

To find the rule of a mapping given only the arrows, use the same difference method. If 1 → 5, 2 → 7, 3 → 9, 4 → 11, the outputs go up by 2, so the rule has 2x, and 5 − 2 = 3, giving $x \to 2 x + 3$ .

4. Working backwards to find an input

Sometimes you know the output and must find the input. For $x \to 2 x + 3$ , which x gives 13? Write it as an equation, 2x + 3 = 13, and undo the operations in reverse order using inverse operations.

▶ Keep the scale balanced: whatever you do to one side, do to the other. Notice we undo the +3 first, then the ×2.

Negative numbers work the same way. For $x \to - 2 x + 5$ with output 45: subtract 5 from both sides to get −2x = 40, then divide by −2 to get x = −20. Check: −2 × (−20) + 5 = 40 + 5 = 45 ✓.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Finding the nth term

Find the rule for the nth term of 7, 11, 15, 19, … and use it to find the 20th term.

Show the solution

Step 1Common difference = 11 − 7 = 4, so the rule has 4n.
Step 2First term − difference = 7 − 4 = 3, so nth term = 4n + 3.
Step 3Check: n = 2 gives 8 + 3 = 11 ✓
Step 420th term = 4 × 20 + 3 = 83.

Answer: 4n + 3; the 20th term is 83

Example 2

A matchstick prediction

Squares in a row use 4, 7, 10, … matchsticks. How many sticks are in the 50th pattern? Which pattern uses exactly 100 sticks?

▶ 1 starting stick plus 3 sticks for each square: 3n + 1.

Show the solution

Step 1Difference 3, and 4 − 3 = 1, so sticks = 3n + 1.
Step 250th pattern: 3 × 50 + 1 = 151 sticks.
Step 33n + 1 = 100 → 3n = 99 → n = 33.
Step 4Check: 3 × 33 + 1 = 100 ✓

Answer: 151 sticks; Pattern 33

Example 3

Using a mapping both ways

For the mapping

x \to 2 x + 3

: (i) map the domain {1, 2, 3, 4, 5}; (ii) find x when the result is 13.

Show the solution

Step 1(i) 1 → 5, 2 → 7, 3 → 9, 4 → 11, 5 → 13.
Step 2(ii) 2x + 3 = 13. Subtract 3: 2x = 10. Divide by 2: x = 5.
Step 3Check: 2 × 5 + 3 = 13 ✓

Answer: (i) {5, 7, 9, 11, 13}; (ii) x = 5

Example 4

A mapping with a negative multiplier

The result of x in the mapping

x \to - 2 x + 5

is 45. Find the value of x.

Show the solution

Step 1Write the equation: −2x + 5 = 45.
Step 2Subtract 5 from both sides: −2x = 40.
Step 3Divide both sides by −2: x = −20.
Step 4Check: −2 × (−20) + 5 = 40 + 5 = 45 ✓

Answer: x = −20

Example 5

A decreasing pattern

Find the nth term of 50, 46, 42, 38, … and decide which term is equal to 2.

Show the solution

Step 1Each term is 4 less, so the difference is −4 and the rule has −4n.
Step 2First term − difference = 50 − (−4) = 54, so nth term = 54 − 4n.
Step 3Check: n = 3 gives 54 − 12 = 42 ✓
Step 454 − 4n = 2 → 4n = 52 → n = 13.

Answer: 54 − 4n; the 13th term is 2

Summary

The common difference d tells you the rule contains dn.
The number to add is first term − d, so the nth term is dn + (first term − d).
Put the term number into the rule to predict any term — never just double an earlier term.
A mapping such as $x \to 2 x + 3$ sends every input to an output with the same rule.
To find an input from an output, form an equation and undo the operations in reverse order.
If solving for n does not give a whole number, the number is not in the pattern.

How many chairs in row 25?

1. From the difference to the rule

2. Rules for matchstick and dot patterns

3. Mappings and their rules

4. Working backwards to find an input

Worked examples

Finding the nth term

A matchstick prediction

Using a mapping both ways

A mapping with a negative multiplier

A decreasing pattern

Summary

Practice: 15 questions