Number Patterns Revision Notes for Grade 10 NSC Matric Mathematics

Describing Sequences

What is a sequence?

A sequence is an ordered list of numbers arranged in a specific pattern. Each individual number in the sequence is called a term. Sequences follow rules that help us predict what comes next.

The diagram shows how we identify and label terms in a sequence. Notice that three dots (...) indicate the sequence continues indefinitely.

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Understanding sequences is fundamental to mathematics because they appear everywhere - from simple counting patterns to complex mathematical relationships. Once you can identify the pattern, you can predict any term in the sequence!

Term notation

When working with sequences, we use special notation to identify the position of each term:

$T_1$ represents the first term
$T_2$ represents the second term
$T_3$ represents the third term
$T_4$ represents the fourth term
$T_n$ represents the nth term (general term)

This notation helps us describe any term's position without having to write out the entire sequence.

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Think of $T_n$ as "the term at position n" - this notation is universal in mathematics and you'll see it in many different contexts beyond just sequences.

Identifying patterns in sequences

Sequences can follow different types of patterns. Here are the main types you need to recognise:

Arithmetic sequences (linear patterns)

These sequences have a constant difference between consecutive terms:

Example: 2, 5, 8, 11, 14, ... (adding 3 each time)
Example: 10, 7, 4, 1, -2, ... (subtracting 3 each time)

Geometric sequences

These sequences have a constant ratio between consecutive terms:

Example: 2, 4, 8, 16, 32, ... (multiplying by 2 each time)
Example: 27, 9, 3, 1, $\frac{1}{3}$ , ... (multiplying by $\frac{1}{3}$ each time)

The focus of this chapter is on arithmetic sequences, also known as linear sequences.

Common difference

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Definition: The common difference is the difference between any term and the term immediately before it. We use the symbol d to represent common difference.

Key Formula: $d = T_n - T_{n-1}$

Critical Point: Always subtract the earlier term from the later term, not the other way around.

How to calculate common difference

For any arithmetic sequence, the common difference is found using: $d = T_n - T_{n-1}$

This means you take any term and subtract the term that comes immediately before it.

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Worked Example: Finding Common Difference

For the sequence 10, 7, 4, 1, ...

Step 1: Apply the formula to consecutive pairs

$d = T_2 - T_1 = 7 - 10 = -3$
$d = T_3 - T_2 = 4 - 7 = -3$
$d = T_4 - T_3 = 1 - 4 = -3$

Step 2: Verify consistency The common difference is d = -3 (negative because the sequence is decreasing).

General formulas for arithmetic sequences

When a sequence has a common difference, we can create a general formula to find any term. The general form is:

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Standard Form: $T_n = a + (n-1)d$

Where:

a is the first term
d is the common difference
n is the position number
$T_n$ is the value of the nth term

Alternative Form: $T_n = dn + c$

Where c is a constant value.

The standard form $T_n = a + (n-1)d$ is often easier to use because you can directly substitute the first term and common difference.

Worked example 1: Table seating pattern

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Worked Example: Table Seating Pattern

Problem: A group arranges tables for studying. With 1 table, 4 people can sit. Each additional table allows 2 more people to sit. Find the pattern.

Step 1: Create a table to see the pattern

Number of tables (n)	People seated	Pattern
1	4	4
2	6	4 + 2(1)
3	8	4 + 2(2)
4	10	4 + 2(3)

Step 2: Identify the pattern

First term: $T_1 = 4$
Common difference: $d = 2$
General formula: $T_n = 4 + 2(n-1)$

Step 3: Verify the formula

$T_1 = 4 + 2(1-1) = 4 + 0 = 4$ ✓
$T_2 = 4 + 2(2-1) = 4 + 2 = 6$ ✓
$T_3 = 4 + 2(3-1) = 4 + 4 = 8$ ✓

Worked example 2: Data pricing pattern

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Worked Example: Data Pricing Pattern

Problem: A data plan costs R120 for 1GB, R135 for 2GB, R150 for 3GB. Find when it's cheaper to buy an unlimited plan costing R520.

Step 1: Identify the pattern

1GB: R120
2GB: R135 (increase of R15)
3GB: R150 (increase of R15)
Common difference: $d = \text{R}15$

Step 2: Find the general formula

First term: $a = \text{R}120$
Formula: $T_n = 120 + 15(n-1)$

Step 3: Find when unlimited is cheaper

Set $T_n = 520$ :
$520 = 120 + 15(n-1)$
$520 = 120 + 15n - 15$
$520 = 105 + 15n$
$415 = 15n$
$n = 27.67$

Therefore, unlimited becomes cheaper from 28GB onwards.

Worked example 3: Decreasing sequence

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Worked Example: Decreasing Sequence

Problem: Find the general formula for the sequence: 6, 1, -4, -9, ...

Step 1: Find the common difference

$d = T_2 - T_1 = 1 - 6 = -5$
Check: $d = T_3 - T_2 = -4 - 1 = -5$ ✓

Step 2: Apply the general formula

First term: $a = 6$
Common difference: $d = -5$
Formula: $T_n = 6 + (n-1)(-5)$
Simplified: $T_n = 6 - 5(n-1)$
Further simplified: $T_n = 11 - 5n$

Step 3: Verify

$T_1 = 11 - 5(1) = 6$ ✓
$T_2 = 11 - 5(2) = 1$ ✓
$T_3 = 11 - 5(3) = -4$ ✓

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Notice how the negative common difference creates a decreasing sequence. This is perfectly normal - arithmetic sequences can increase, decrease, or even stay constant (when d = 0).

Key exam tips

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Essential Points to Remember:

Always check your common difference by testing at least two consecutive pairs
Remember the correct order when calculating $d = T_n - T_{n-1}$ (later term minus earlier term)
Linear sequences are another name for arithmetic sequences
Verify your general formula by substituting small values of n
Be careful with negative common differences - they create decreasing sequences
Read word problems carefully to identify what represents n and what represents $T_n$

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Common Mistakes to Avoid:

Calculating common difference in the wrong order (don't do $T_{n-1} - T_n$ )
Forgetting to verify your formula with the original terms
Mixing up position number (n) with term value ( $T_n$ ) in word problems
Assuming all sequences are arithmetic - always check for constant differences first

Number Patterns (Grade 10 NSC Matric Mathematics): Revision Notes

Describing Sequences

What is a sequence?

Term notation

Identifying patterns in sequences

Arithmetic sequences (linear patterns)

Geometric sequences

Common difference

How to calculate common difference

General formulas for arithmetic sequences

Worked example 1: Table seating pattern

Worked example 2: Data pricing pattern

Worked example 3: Decreasing sequence

Key exam tips

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