Arithmetic Sequences Revision Notes for Grade 12 NSC Matric Mathematics

Arithmetic Sequences

What is an arithmetic sequence?

An arithmetic sequence is a special type of number pattern where each term is found by adding the same constant value to the previous term. This constant value is called the common difference.

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Definition: An arithmetic sequence is an ordered set of numbers where consecutive terms are calculated by adding a constant value to the previous term.

For example, consider the sequence: 3, 0, -3, -6, -9, ...

In this sequence, we add -3 to each term to get the next term. The common difference (d) is -3.

The key components of an arithmetic sequence are:

a = the first term
d = the common difference (the value added each time)
n = the position of a term in the sequence
Tn = the value of the nth term

The general term formula

To find any term in an arithmetic sequence without listing all previous terms, we use the general term formula.

Starting with the first term a, we can build the sequence:

$T_1 = a$
$T_2 = a + d$
$T_3 = a + 2d$
$T_4 = a + 3d$

Notice the pattern: each term equals the first term plus the common difference multiplied by $(n-1)$ .

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General Term Formula: $T_n = a + (n - 1)d$

Where:

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

How to test for arithmetic sequences

To determine if a sequence is arithmetic, check whether the difference between consecutive terms is constant:

Test: $d = T_2 - T_1 = T_3 - T_2 = T_4 - T_3 = ... = T_n - T_{n-1}$

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If all these differences are equal, the sequence is arithmetic. If not, it's not an arithmetic sequence.

Arithmetic mean

The arithmetic mean between two numbers is the average of those numbers. When you place the arithmetic mean between two numbers, you create an arithmetic sequence of three terms.

Formula: Arithmetic mean = $\frac{\text{first number} + \text{second number}}{2}$

For example, the arithmetic mean between 7 and 17 is: Arithmetic mean = $\frac{7 + 17}{2} = \frac{24}{2} = 12$

So 7, 12, 17 forms an arithmetic sequence with $d = 5$ .

Graphical representation of arithmetic sequences

When you plot the terms of an arithmetic sequence on a coordinate graph, they form a straight line. This is why arithmetic sequences are also called linear sequences.

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The graph shows that:

The points lie on a straight line
The gradient of the line equals the common difference (d)
The general formula $T_n = a + (n-1)d$ has the same form as $y = mx + c$

This linear relationship makes arithmetic sequences easy to recognise when graphed.

Worked example

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Worked Example: Analysing an Arithmetic Sequence

Question: Given the sequence -15, -11, -7, ..., 173

Is this an arithmetic sequence?
Find the formula for the general term
Determine the number of terms in the sequence

Solution:

Step 1: Check for common difference
$T_2 - T_1 = -11 - (-15) = 4$
$T_3 - T_2 = -7 - (-11) = 4$

Since the difference is constant, this is an arithmetic sequence with $d = 4$ .

Step 2: Find the general term formula
Using $T_n = a + (n - 1)d$ where $a = -15$ and $d = 4$ :

$T_n = -15 + (n - 1)(4)$
$T_n = -15 + 4n - 4$
$T_n = 4n - 19$

Step 3: Find the number of terms
The last term is 173, so:
$173 = 4n - 19$
$192 = 4n$
$n = 48$

Therefore, there are 48 terms in the sequence.

Key characteristics of arithmetic sequences

Understanding these key properties helps identify and work with arithmetic sequences:

Constant difference: The difference between consecutive terms is always the same
Linear growth: Terms increase or decrease at a steady rate
Straight line graph: When plotted, the points form a straight line
General formula: Always in the form $T_n = a + (n-1)d$
Predictable pattern: Any term can be calculated without finding all previous terms

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Key Points to Remember:

An arithmetic sequence has a constant difference between consecutive terms
The general term formula is $T_n = a + (n - 1)d$ , where $a$ is the first term and $d$ is the common difference
To test if a sequence is arithmetic, check that all consecutive differences are equal
Arithmetic sequences form straight lines when graphed, making them linear sequences
The arithmetic mean of two numbers creates the middle term of a three-term arithmetic sequence

Arithmetic Sequences (Grade 12 NSC Matric Mathematics): Revision Notes