Extending Numerical Patterns

A numerical pattern is an ordered list of numbers that follows a rule. When the numbers are arranged in order, the pattern is often called a sequence.

For example:

4, 7, 10, 13, 16, ...

Each number in the sequence is called a term.

The rule for this sequence is:

Add 3 each time.

Understanding the rule allows us to extend the pattern, predict later terms, and describe how the numbers change.

Looking for the Rule

When you are given a numerical pattern, begin by comparing neighbouring terms.

Consider:

5, 9, 13, 17, 21, ...

Calculate the changes:

9 − 5 = 4

13 − 9 = 4

17 − 13 = 4

21 − 17 = 4

The same number is added each time.

Therefore, the rule is:

Start at 5 and add 4 each time.

The next terms are:

25, 29, 33, ...

Increasing Patterns

An increasing pattern produces larger values as the sequence continues.

For example:

12, 17, 22, 27, 32, ...

The rule is:

Add 5.

Therefore:

32 + 5 = 37

37 + 5 = 42

The next two terms are:

37, 42

Decreasing Patterns

Patterns can also decrease.

Consider:

50, 44, 38, 32, 26, ...

The change is:

−6

The rule can be described as:

Start at 50 and subtract 6 each time.

The next terms are:

26 − 6 = 20

20 − 6 = 14

So:

20, 14

Using a Difference Table

A difference table can make a pattern easier to see.

Consider:

7, 12, 17, 22, 27

Because the difference is always +5, the pattern has a constant difference.

A sequence with a constant difference is called an arithmetic sequence.

Multiplication Patterns

Not every pattern involves addition or subtraction.

Consider:

3, 6, 12, 24, 48, ...

Compare consecutive terms:

6 ÷ 3 = 2

12 ÷ 6 = 2

24 ÷ 12 = 2

48 ÷ 24 = 2

The rule is:

Multiply by 2 each time.

The next terms are:

48 × 2 = 96

96 × 2 = 192

So:

96, 192

Division Patterns

A pattern can also involve repeated division.

For example:

160, 80, 40, 20, 10, ...

Each term is divided by 2.

The next terms are:

10 ÷ 2 = 5

5 ÷ 2 = 2.5

Therefore:

5, 2.5

Describing a Pattern Clearly

A good pattern rule should include enough information for another person to recreate the sequence.

For:

8, 13, 18, 23, ...

saying:

"Add 5"

describes how the pattern changes.

An even clearer description is:

"Start at 8 and add 5 to get each new term."

This tells us both the starting value and the operation.

Recursive Rules

A rule that tells us how to find a term from the previous term is called a recursive rule.

For example:

10, 14, 18, 22, ...

The recursive rule is:

Start with 10 and add 4 to the previous term.

We can write:

next term = previous term + 4

Recursive rules are especially useful when extending a pattern one term at a time.

Predicting Terms Farther Along

Extending a sequence one term at a time works well when we only need the next few terms.

But suppose we have:

5, 8, 11, 14, 17, ...

and want the 50th term.

Adding 3 repeatedly 45 more times would be inefficient.

Instead, we can find a rule connecting the term number directly to the term value.

Term Numbers

Every term has a position.

For the sequence:

5, 8, 11, 14, 17, ...

The term number is usually represented by n.

We want an equation that produces the correct value for any n.

Finding a General Rule

The sequence:

5, 8, 11, 14, 17, ...

increases by 3.

Start with:

3n

This produces:

3, 6, 9, 12, 15, ...

But our sequence is:

5, 8, 11, 14, 17, ...

Each required value is 2 greater.

Therefore:

term = 3n + 2

We can write:

Tₙ = 3n + 2

where Tₙ means the value of term number n.

Check:

For n = 1:

T₁ = 3(1) + 2 = 5

For n = 4:

T₄ = 3(4) + 2 = 14

The rule works.

Predicting a Farther Term

Using:

Tₙ = 3n + 2

find the 50th term.

Substitute:

n = 50

T₅₀ = 3(50) + 2

T₅₀ = 150 + 2

T₅₀ = 152

We can find the 50th term without writing all 50 terms.

Another Example

Consider:

11, 18, 25, 32, 39, ...

The constant difference is:

+7

Start with:

7n

This gives:

7, 14, 21, 28, 35, ...

Compare with the actual sequence:

11, 18, 25, 32, 39, ...

Each value is 4 greater.

Therefore:

Tₙ = 7n + 4

Find the 20th term:

T₂₀ = 7(20) + 4

T₂₀ = 140 + 4

T₂₀ = 144

Arithmetic Sequence Formula

For an arithmetic sequence, we can also use:

Tₙ = a + (n − 1)d

where:

• Tₙ = nth term
• a = first term
• n = term number
• d = common difference

Example

Consider:

6, 10, 14, 18, ...

Here:

a = 6

d = 4

Therefore:

Tₙ = 6 + (n − 1)(4)

To find the 25th term:

T₂₅ = 6 + (25 − 1)(4)

T₂₅ = 6 + 96

T₂₅ = 102

Patterns with Negative Numbers

Patterns can cross zero and continue into negative numbers.

For example:

15, 10, 5, 0, −5, −10, ...

The rule is:

Subtract 5 each time.

The next terms are:

−15, −20

A negative value does not mean the pattern has ended.

Patterns with Decimals

The same strategies work with decimals.

Consider:

1.5, 2.0, 2.5, 3.0, 3.5, ...

The constant difference is:

+0.5

The next terms are:

4.0, 4.5

Patterns with Fractions

Consider:

1/2, 1, 3/2, 2, 5/2, ...

Each term increases by:

1/2

The next terms are:

3, 7/2

The numbers may look different, but the same pattern strategies apply.

Patterns with Changing Differences

Some sequences do not have a constant difference.

Consider:

1, 4, 9, 16, 25, ...

Calculate the differences:

4 − 1 = 3

9 − 4 = 5

16 − 9 = 7

25 − 16 = 9

The differences are:

3, 5, 7, 9, ...

The differences themselves form a pattern.

The original sequence consists of square numbers:

1², 2², 3², 4², 5², ...

Therefore:

Tₙ = n²

The next term is:

6² = 36

Another Changing Pattern

Consider:

2, 5, 10, 17, 26, ...

Find the differences:

5 − 2 = 3

10 − 5 = 5

17 − 10 = 7

26 − 17 = 9

Again, the differences are:

3, 5, 7, 9, ...

Compare the terms with square numbers:

n²:

1, 4, 9, 16, 25, ...

Our sequence:

2, 5, 10, 17, 26, ...

Each term is one greater.

Therefore:

Tₙ = n² + 1

The next term is:

6² + 1 = 37

Alternating Patterns

Some sequences alternate between different operations or values.

For example:

3, 8, 4, 9, 5, 10, 6, ...

One way to understand this is to separate the sequence into two patterns.

Odd-position terms:

3, 4, 5, 6, ...

Even-position terms:

8, 9, 10, ...

Both increase by 1.

Alternating patterns often require looking at the odd and even terms separately.

Multi-Step Rules

Some patterns use more than one operation.

Consider:

2, 5, 11, 23, 47, ...

Look at how each term changes:

2 × 2 + 1 = 5

5 × 2 + 1 = 11

11 × 2 + 1 = 23

23 × 2 + 1 = 47

The rule is:

Multiply by 2, then add 1.

The next term is:

47 × 2 + 1 = 95

Finding Missing Terms

Sometimes a sequence contains missing values.

For example:

12, __, 22, 27, 32

The visible values suggest a constant difference of +5.

Therefore:

12 + 5 = 17

So:

12, 17, 22, 27, 32

Another Example

3, 9, __, 81, 243

Each term is multiplied by 3.

Therefore:

9 × 3 = 27

The missing term is 27.

Creating Your Own Numerical Pattern

Creating a pattern requires choosing:

1. a starting value
2. a rule
3. several terms that follow the rule

Example

Starting value:

7

Rule:

Multiply by 2 and subtract 1

Generate the sequence:

7

7 × 2 − 1 = 13

13 × 2 − 1 = 25

25 × 2 − 1 = 49

49 × 2 − 1 = 97

So the pattern is:

7, 13, 25, 49, 97, ...

A clear description would be:

Start at 7. Multiply each term by 2 and subtract 1 to obtain the next term.

More Than One Possible Pattern

A short list of numbers can sometimes fit more than one rule.

Consider:

2, 4, 6, ...

The most obvious rule is:

Add 2 each time.

This predicts:

8, 10, ...

But with only three terms, more complicated rules could also be invented.

For school pattern problems, we normally look for the simplest reasonable rule that fits all the given terms.

The more terms we are given, the more evidence we have for the intended pattern.

Real-World Connections

Numerical patterns appear in many situations.

Saving Money

If you save $25 each week, your total savings might follow:

25, 50, 75, 100, 125, ...

Seating

If each new row contains 4 more seats than the previous row:

20, 24, 28, 32, ...

Population Growth

If a population doubles during each time interval:

100, 200, 400, 800, ...

Depreciation

If an object's value is repeatedly multiplied by 0.8:

1000, 800, 640, 512, ...

Recognizing the pattern allows us to make predictions.

A Strategy for Investigating Patterns

When you encounter a new sequence, ask:

Are the terms increasing or decreasing?

Then check:

Is the same amount being added or subtracted?

If not:

Is each term multiplied or divided by the same number?

If not:

Do the differences themselves form a pattern?

Then consider:

• square numbers
• cube numbers
• alternating patterns
• repeated multi-step rules
• relationships between term number and term value

Finally, test your proposed rule against every term given.

Common Mistakes

Looking only at the first two terms

A rule should fit all the terms provided.

Confusing addition with multiplication

For:

5, 10, 20, 40, ...

the difference is not constant. The terms are multiplied by 2.

Forgetting the starting value

"Add 4" does not completely define a sequence unless the starting term is known.

Assuming every pattern has a constant difference

Some patterns involve multiplication, changing differences, squares, or alternating rules.

Making an arithmetic error when extending a correct pattern

After finding the rule, apply it carefully to each new term.

Extending a long sequence one term at a time

For distant terms in arithmetic sequences, an nth-term rule is much more efficient.

Key Terms

Numerical pattern — an ordered collection of numbers generated according to a rule.

Sequence — an ordered list of terms.

Term — an individual number in a sequence.

Term number — the position of a term within a sequence.

Pattern rule — the instruction or relationship used to generate the terms.

Recursive rule — a rule that uses one term to determine the next term.

Arithmetic sequence — a sequence with a constant difference between consecutive terms.

Common difference — the constant amount added or subtracted in an arithmetic sequence.

Geometric pattern — a pattern in which terms are repeatedly multiplied by the same factor.

General rule — an equation connecting the term number to the term value.

Nth term — a general expression that gives the value of any term based on its position n.

Alternating pattern — a sequence in which two or more patterns alternate.

Key Takeaways

• Numerical patterns follow identifiable rules.
• Each number in a sequence is called a term.
• Comparing consecutive terms can reveal how a pattern changes.
• Constant addition or subtraction produces an arithmetic sequence.
• Some patterns involve repeated multiplication or division.
• Some patterns have changing differences or multi-step rules.
• Recursive rules describe how to generate the next term.
• An nth-term rule allows us to predict terms much farther along a sequence.
• For an arithmetic sequence, Tₙ = a + (n − 1)d.
• A proposed rule should fit all the given terms.
• Numerical patterns can involve positive numbers, negative numbers, fractions, and decimals.
• Creating your own pattern requires both a starting value and a clear rule.

Check Your Understanding

1. Find the next three terms:

   6, 10, 14, 18, ...

2. Describe the rule for:

   45, 38, 31, 24, ...

3. Find the next two terms:

   5, 15, 45, 135, ...

4. Find the missing term:

   18, 24, __, 36, 42

5. Find the next two terms:

   1, 4, 9, 16, 25, ...

6. Describe the rule for:

   4, 9, 19, 39, 79, ...

7. The sequence is:

   8, 13, 18, 23, ...

   Find the 20th term.

8. Find an nth-term rule for:

   4, 7, 10, 13, 16, ...

9. Use your rule from Question 8 to find the 50th term.
10. Determine the next term:

3, 6, 4, 8, 5, 10, 6, ...

11. A sequence begins at 12. Each new term is found by multiplying the previous term by 2 and subtracting 3. Write the first six terms.
12. Create your own numerical pattern containing at least six terms. State the starting value, describe the rule clearly, and predict the tenth term.