Arithmetic Sequences

Expert reviewed • 21 July 2024 • 5 minute read

HSC Maths Advanced Syllabus

recognise and use the recursive definition of an arithmetic sequence: $T_n = T_{n-1} + d$ , $T_1 = a$
establish and use the formula for the $n^{th}$ term (where $n$ is a positive integer) of an arithmetic sequence: $T_n = a + (n - 1)d$ , where $a$ is the first term and $d$ is the common difference, and recognise its linear nature

Note: ‘Arithmetic Sequence’ is a term synonymous with ‘Arithmetic Progression’ (AP is the shorthand).

What is an Arithmetic Sequence?

An arithmetic sequence or AP for short, is a sequence in which the difference between consecutive terms is constant. This constant difference is known as the "common difference" represented by the letter d

The sequence progresses by adding (or subtracting, if d is negative) the common difference d to each term to get the next term.

Finding the nth term of an AP

The formula to find the nth term of an arithmetic sequence is given by:

T_n=a+(n-1)d

Where $a$ is the first term in the sequence $(T_1)$ and $d$ is the common difference.

Practice Question 1

Find a formula for the nth term of the arithmetic sequence: $3,6,9,12,...$

From the information given, we know that a = 3 and d = 3. Thus,

T_n = 3+(n-1)3\\=3+3n-3\\=3n

Practice Question 2

How many terms does the arithmetic series $3,6,9,...,45$ contain?

From the previous question, we know that the formula for the nth term of this sequence is:

$T_n = 3n$

Thus, to find n we must equate $T_n$ with the last term of the sequence.

T_n = 45\\3n=45\\n=15

Thus, there are 15 terms in the sequence

Can an AP be formed using 3 numbers?

If the terms a, b and c are used in a sequence, as shown below:

a,b,c

We can use the following formulas to determine if the sequence is an AP, and if it is they can be used to find missing terms.

b-a = c-b\qquad \quad b=\frac{1}{2}(a+c)

Practice Question 3

Find the value of b, if the terms $15,b,25$ form an AP

b = \frac{1}{2}(15+25)\\=\frac{1}{2}(40)\\=20

Introduction to Sequences & Series

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Geometric Sequences

Return to Module 1: Sequences and Series