Arithmetic Sequences Revision Notes for HSC SSCE Mathematics Advanced

Arithmetic Sequences

What is an arithmetic sequence?

An arithmetic sequence is a pattern of numbers where each term differs from the previous term by the same constant amount. This constant amount is called the common difference.

For example, consider the sequence:

$3, 13, 23, 33, 43, 53, 63, 73, 83, 93, \ldots$

In this sequence, each term is $10$ more than the previous term. The common difference is $d = 10$ . All terms can be generated from the first term ( $3$ ) by repeatedly adding this common difference.

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Key terminology:

An arithmetic sequence is also called an AP (arithmetic progression)
The difference between successive terms always means: current term minus previous term

Formal definition

A sequence $T_n$ is called an arithmetic sequence if the difference between any term and the previous term is constant.

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Mathematical Definition:

A sequence $T_n$ is an arithmetic sequence if:

$T_n - T_{n-1} = d \text{ for all } n \geq 2$

where $d$ is a constant called the common difference.

Key properties:

The difference between successive terms: $\text{difference} = T_n - T_{n-1}$ where $n \geq 2$
Terms can be generated by repeated addition: $T_n = T_{n-1} + d$ for all $n \geq 2$

Testing if a sequence is an AP

To determine whether a sequence is arithmetic, calculate the differences between consecutive terms. If all differences are equal, the sequence is an AP.

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Worked Example: Testing for Arithmetic Sequences

Test whether each sequence is an AP. If it is, find the first term $a$ and common difference $d$ .

a) $46, 43, 40, 37, \ldots$

Calculate consecutive differences:

T_2 - T_1 = 43 - 46 = -3 T_3 - T_2 = 40 - 43 = -3 T_4 - T_3 = 37 - 40 = -3

Since all differences equal $-3$ , this is an AP with $a = 46$ and $d = -3$ .

b) $1, 4, 9, 16, \ldots$

Calculate consecutive differences:

T_2 - T_1 = 4 - 1 = 3 T_3 - T_2 = 9 - 4 = 5 T_4 - T_3 = 16 - 9 = 7

The differences are not all the same ( $3$ , $5$ , $7$ are different), so this sequence is not an AP.

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Exam tip: Always check at least three consecutive differences to confirm a pattern. Remember that the differences can be negative (decreasing sequence) or positive (increasing sequence).

The nth term formula

For an AP with first term $a$ and common difference $d$ , the first few terms follow a clear pattern:

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

Notice that each term is the first term plus a multiple of the common difference. The multiplier is always one less than the term number.

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Formula for the nth term:

$T_n = a + (n - 1)d$

where:

$a$ is the first term
$d$ is the common difference
$n$ is the term number

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Worked Example: Finding Terms Using the Formula

Write out the first five terms and calculate the 20th term of the AP with:

a) $a = 2$ and $d = 5$

First five terms: $2, 7, 12, 17, 22, \ldots$

For the 20th term:

T_{20} = a + 19d T_{20} = 2 + 19 \times 5 T_{20} = 2 + 95 T_{20} = 97

b) $a = 20$ and $d = -3$

First five terms: $20, 17, 14, 11, 8, \ldots$

For the 20th term:

T_{20} = a + 19d T_{20} = 20 + 19 \times (-3) T_{20} = 20 - 57 T_{20} = -37

Finding the nth term formula

When given the first few terms of an AP, you can find a formula for the nth term by identifying the first term and common difference.

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Worked Example: Deriving the nth Term Formula

a) Find a formula for the nth term of the sequence $26, 35, 44, 53, \ldots$

First, identify that this is an AP with $a = 26$ and $d = 9$ .

Using the formula:

T_n = a + (n - 1)d T_n = 26 + 9(n - 1) T_n = 26 + 9n - 9 T_n = 17 + 9n

b) How many terms are there in the sequence $26, 35, 44, 53, \ldots, 917$ ?

We need to find $n$ when $T_n = 917$ .

Using our formula from part (a):

17 + 9n = 917 9n = 900 n = 100

Therefore, there are 100 terms in the sequence.

Solving problems with arithmetic sequences

The nth term formula allows us to solve various types of problems by forming and solving equations.

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Worked Example: Finding the First Negative Term

Show that the sequence $200, 193, 186, \ldots$ is an AP, find a formula for the nth term, and find the first negative term.

Solution:

a) Test if it's an AP:

T_2 - T_1 = 193 - 200 = -7 T_3 - T_2 = 186 - 193 = -7

Since the differences are equal, this is an AP with $a = 200$ and $d = -7$ .

b) Find the formula:

T_n = 200 - 7(n - 1) T_n = 200 - 7n + 7 T_n = 207 - 7n

c) Find the first negative term:

We need $T_n < 0$ :

207 - 7n < 0 207 < 7n n > 29\frac{4}{7}

Since $n$ must be a positive integer, the first negative term is T₃₀.

Calculating: $T_{30} = 207 - 7(30) = 207 - 210 = -3$

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Worked Example: Finding Unknown Values

The first term of an AP is $105$ and the 10th term is $6$ . Find the common difference and write out the first five terms.

Solution:

We know that $T_1 = 105$ , so $a = 105$ ... (1)

We also know that $T_{10} = 6$

Using the formula for the 10th term:

$a + 9d = 6 \text{ ... (2)}$

Substituting equation (1) into equation (2):

105 + 9d = 6 9d = -99 d = -11

Therefore, the common difference is d = -11 and the first five terms are:

$105, 94, 83, 72, 61, \ldots$

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Exam tip: When you know two terms in an AP, you can set up two equations and solve for the unknowns $a$ and $d$ .

Arithmetic sequences and linear functions

Arithmetic sequences are closely related to linear functions. When you substitute positive integers into a linear function, you generate an arithmetic sequence.

Example:

Consider the linear function $f(x) = 30 - 8x$ .

Substituting positive integers for $x$ :

$x$	$1$	$2$	$3$	$4$	$5$
$f(x)$	$22$	$14$	$6$	$-2$	$-10$

This generates the arithmetic sequence: $22, 14, 6, -2, -10, \ldots$

The formula for the nth term of this AP is:

$T_n = 22 - 8(n - 1) = 30 - 8n$

Notice that this is the same as the original linear function, with the variable $x$ replaced by $n$ .

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Key insight:

An arithmetic sequence is a function whose domain is the set of positive integers. The nth term formula is equivalent to a linear function, just using different notation.

Every arithmetic sequence can be generated by substituting positive integers into an appropriate linear function.

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Key Points to Remember:

An arithmetic sequence (AP) has a constant difference between consecutive terms
The common difference is $d = T_n - T_{n-1}$
The nth term formula is $T_n = a + (n-1)d$ where $a$ is the first term
To test if a sequence is an AP, check that all consecutive differences are equal
Arithmetic sequences are discrete versions of linear functions - they follow the same pattern but only for positive integer values

Arithmetic Sequences (HSC SSCE Mathematics Advanced): Revision Notes