Summary Revision Notes for Grade 11 NSC Matric Mathematics

Summary

This section covers the essential concepts of number patterns, specifically focusing on linear and quadratic sequences. Understanding these patterns is crucial for identifying sequence types and finding general terms in mathematical sequences.

What is a sequence?

A sequence is an ordered list of numbers that follow a specific pattern. Each number in the sequence is called a term.

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The general term of a sequence is represented by $T_n$ , where $n$ indicates the position of the term in the sequence. For example, $T_1$ is the first term, $T_2$ is the second term, and so on.

Successive terms (also called consecutive terms) are terms that appear one after another in a sequence. These are any two terms that are next to each other in the sequence order.

Linear sequences

A linear sequence is a sequence where there is a constant difference between any two successive terms. This constant difference is called the common difference and is represented by the letter $d$ .

Finding the common difference

The formula for calculating the common difference is:

$d = T_n - T_{n-1}$

This means you subtract any term from the term that comes immediately after it.

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Key properties of linear sequences:

The first differences are always constant
When you plot the terms on a graph, they form a straight line
The pattern increases or decreases at a steady rate

Quadratic sequences

A quadratic sequence is a sequence where the second differences between successive terms are constant. Unlike linear sequences, the first differences are not constant, but when you find the differences of the differences (second differences), these remain constant.

General term of a quadratic sequence

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The general term for a quadratic sequence follows the formula:

$T_n = an^2 + bn + c$

where $a$ , $b$ , and $c$ are constants, and $a \neq 0$ .

Understanding differences in quadratic sequences

The diagram above illustrates how differences work in quadratic sequences. Here's what happens:

First difference: The difference between consecutive terms
Second difference: The difference between consecutive first differences

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For any quadratic sequence:

The second differences are always constant
The second difference equals $2a$ (where $a$ is the coefficient of $n^2$ in the general term)
This constant second difference is the key identifying feature

Worked examples

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Worked Example: Finding the first five terms of a quadratic sequence

Given: $T_n = n^2 + 2n + 1$

Find the first five terms:

$T_1 = 1^2 + 2(1) + 1 = 1 + 2 + 1 = 4$
$T_2 = 2^2 + 2(2) + 1 = 4 + 4 + 1 = 9$
$T_3 = 3^2 + 2(3) + 1 = 9 + 6 + 1 = 16$
$T_4 = 4^2 + 2(4) + 1 = 16 + 8 + 1 = 25$
$T_5 = 5^2 + 2(5) + 1 = 25 + 10 + 1 = 36$

The first five terms are: 4, 9, 16, 25, 36

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Worked Example: Identifying a linear sequence

Consider the sequence: 6, 9, 14, 21, 30, ...

Step 1: Find the first differences

$9 - 6 = 3$
$14 - 9 = 5$
$21 - 14 = 7$
$30 - 21 = 9$

Step 2: Check if differences are constant The first differences (3, 5, 7, 9) are not constant.

Step 3: Find the second differences

$5 - 3 = 2$
$7 - 5 = 2$
$9 - 7 = 2$

Since the second differences are constant (all equal 2), this is a quadratic sequence.

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Worked Example: Identifying sequence types

For the sequence: 1, 7, 17, 31, 49, ...

First differences: 6, 10, 14, 18

Second differences: 4, 4, 4

Since the second differences are constant, this is a quadratic sequence.

How to determine sequence type

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Follow this systematic approach:

Calculate first differences between consecutive terms
If first differences are constant → Linear sequence
If first differences are not constant → Calculate second differences
If second differences are constant → Quadratic sequence
If neither first nor second differences are constant → Neither linear nor quadratic

Exam tips

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Essential exam strategies:

Always work systematically: Start with first differences, then move to second differences if needed
Show your working clearly: Examiners want to see your difference calculations
Check your pattern: Make sure the differences you find are truly constant
Verify answers: Substitute values back into formulas to double-check your work
Remember the key formulas: $d = T_n - T_{n-1}$ for linear sequences, and $T_n = an^2 + bn + c$ for quadratic sequences

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Key Points to Remember:

General term $T_n$ represents any term in a sequence based on its position $n$
Linear sequences have constant first differences and increase/decrease at a steady rate
Quadratic sequences have constant second differences and contain an $n^2$ term in their general formula
Use the common difference formula $d = T_n - T_{n-1}$ to identify linear patterns
Always work through differences systematically: first differences first, then second differences if needed

Summary (Grade 11 NSC Matric Mathematics): Revision Notes

Summary

What is a sequence?

Linear sequences

Finding the common difference

Quadratic sequences

General term of a quadratic sequence

Understanding differences in quadratic sequences

Worked examples

How to determine sequence type

Exam tips

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