Number Patterns (Arithmetic Sequences)

Master linear patterns, common difference, and nth term formulas

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and focuses on identifying, describing, and generalizing linear number patterns. Use the interactive graph below to visualize arithmetic sequences as linear functions.

Arithmetic Sequence Graph: T(n) = a + (n-1)d

Graph Controls

Adjust Sequence Parameters

a (first term)

d (common difference)

Current Sequence: T(n) = 3 + (n-1)4 = 4n - 1

First 6 terms: 3, 7, 11, 15, 19, 23

Hover over graph to see coordinates: n = 1, T = 3

Learning Outcomes

Identify and describe linear number patterns
Calculate the common difference in arithmetic sequences
Find the nth term (general term) of a sequence
Determine specific terms using the nth term formula
Solve real-world problems involving number patterns
Distinguish between linear and non-linear patterns
Apply pattern recognition to mathematical problems

1. Introduction to Linear Number Patterns

Linear Pattern

Also called arithmetic sequence

Constant difference between terms
Forms a straight line when graphed
Example: 2, 5, 8, 11, 14, ...
Difference is always +3

2 5 8 11 14

Common Difference (d)

d = T₂ - T₁ = T₃ - T₂ = ...

Always constant in linear patterns

Can be positive, negative, or zero

First Term (a)

The starting value of the sequence

Denoted as T₁ or simply 'a'

Example: In 3, 7, 11, 15, ... a = 3

Identifying Linear Patterns

Example

Determine if 3, 7, 11, 15, ... is a linear pattern.

Solution

7 - 3 = 4

11 - 7 = 4

15 - 11 = 4

All differences = 4 → Linear pattern ✓

Quiz 1 - Linear Patterns

What is the common difference in 5, 9, 13, 17, ...

A) 3

B) 4

C) 5

D) 2

2. The nth Term Formula

nth Term Formula
Tₙ = a + (n - 1)d
where:
Tₙ = nth term
a = first term
n = term number
d = common difference

Finding the nth Term

Problem

Find the nth term for: 3, 7, 11, 15, ...

Solution

a = 3, d = 4

Tₙ = 3 + (n - 1)4

Tₙ = 3 + 4n - 4 = 4n - 1

Check: T₁ = 4(1) - 1 = 3 ✓

Quiz 2 - nth Term

For sequence 2, 5, 8, 11, ..., what is Tₙ

A) 3n - 1

B) 3n + 1

C) 2n + 3

D) 4n - 2

3. Finding Specific Terms

Example 1

Using Tₙ = 4n - 1, find the 10th term.

T₁₀ = 4(10) - 1 = 39

Example 2

Sequence: 2, 5, 8, 11, ... Find 25th term.

T₂₅ = 2 + 24×3 = 74

Example 3

a = 8, d = -3. Find 12th term.

T₁₂ = 8 + 11(-3) = -25

4. Real-World Applications

Brick Stack Problem

Problem

Brick stack: bottom row 20 bricks, next 18, next 16, etc. How many in 8th row

Solution

Sequence: 20, 18, 16, ... (a=20, d=-2)

Tₙ = 20 + (n-1)(-2) = 22 - 2n

T₈ = 22 - 16 = 6 bricks

Seating Arrangement

Problem

Theatre: 1st row 10 seats, 2nd row 12, 3rd row 14. How many in 15th row

Solution

a=10, d=2, T₁₅ = 10 + 14×2 = 38 seats

Quiz 3 - Applications

Save R50 first week, R55 second, R60 third. How much in week 20

A) R145

B) R150

C) R140

D) R155

5. Finding Term Number (n)

Finding Position of a Term

Problem

In sequence 5, 9, 13, 17, ..., which term is 81

Solution

a=5, d=4, Tₙ = 4n + 1

4n + 1 = 81 → 4n = 80 → n = 20

81 is the 20th term

6. Different Types of Patterns

Increasing Patterns

Positive d: 4, 7, 10, 13, ...

Decreasing Patterns

Negative d: 20, 17, 14, 11, ...

Constant Patterns

d = 0: 5, 5, 5, 5, ...

Practice & Assess

Test your knowledge with these interactive games.

Match - Pattern to Formula

3, 7, 11, 15

4n - 1

2, 5, 8, 11

3n - 1

5, 9, 13, 17

4n + 1

1, 4, 7, 10

3n - 2

Fill - nth Term Formula

Tₙ = a + (n - 1) × ___

7. Common Misconceptions

Error 1

n vs Tₙ: Confusing term number with term value

Error 2

Wrong d calculation: d = next term - current term

Error 3

Formula errors: Tₙ = a + n×d (wrong!)

Correct: Tₙ = a + (n-1)d

Practice Questions

Sequence: 6, 11, 16, 21, ... Find T₈.

Find nth term: 4, 1, -2, -5, ...

In sequence 7, 12, 17, ..., which term is 87

Formula Summary

Key Formulas

d = T₂ - T₁
Tₙ = a + (n - 1)d
n = (Tₙ - a)/d + 1

Important Notes

n must be positive integer
d can be +, -, or 0
Linear = constant difference

Key Terms

Sequence Pattern Linear Arithmetic Common difference First term nth term Term number General term Increasing Decreasing Constant

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