Geometric Sequences

Expert reviewed • 21 July 2024 • 6 minute read

HSC Maths Advanced Syllabus

recognise and use the recursive definition of a geometric sequence: $Tn = rT_{n-1}, T_1=a$
establish and use the formula for the $n$ th term of a geometric sequence: $T_n=ar^{n-1}$ , where 𝑎 is the first term, 𝑟 is the common ratio and 𝑛 is a positive integer, and recognise its exponential nature

What is a Geometric Sequence?

A geometric sequence or GP for short, is a sequence of numbers where each term after the first, is found by multiplying the previous term by a fixed ratio. This ratio is known as the "common ratio," and is represented by the letter r. A sequence is therefore noted as geometric progression, if the following condition is fulfilled:

r=\frac{T_n}{T_{n-1}}

Finding the nth Term of a GP

The formula to find the $n$ th **term of a geometric sequence is given by:

T_n = ar^{n-1}

Where $a$ is the first term in the sequence $T_1$ and $r$ is the common ratio.

Practice Question 1

Find the formula for the $n$ th term of a geometric sequence, that starts at 2 and has a common ratio of 3.

From the given information we know that $a=2$ and $r=3$ . Thus,

T_n=2\times 3^{n-1}

Practice Question 2

Determine if the values 54 and 105 are terms of the sequence, defined by the equation found in the previous question.

From the information given, we know that $T_n=2\times 3^{n-1}$ :

Thus, to determine if the terms 54 and 105 are part of this sequence, we must equate them with the formula to find n:

54 = 2\times3^{n-1} \qquad \qquad 105 = 2\times3^{n-1}\\27=3^{n-1} \qquad \qquad 52.5 = 3^{n-1}\\n-1=3\qquad\qquad n-1=\sqrt[3]{52.5}\\n=4 \qquad\qquad n=\sqrt[3]{52.5}+1

$\therefore$ We can see that 54 is the 4th term of the sequence, and 105 is not a term (as the answer for $n$ **is not an integer)

Can a GP be Formed Using 3 numbers?

e,f,g

We can use the following formulas to determine if the sequence is an GP, and to find any missing variables.

\frac{f}{e}=\frac{g}{f}\qquad\qquad f^2=eg

Practice Question 3

Determine if the following sequence is a GP: $5,10,20$

From the information given, we know that $e=5, f=10 , g=20$ .

Substituting in values:

10^2=5\times20\\100=100

$\therefore$ By using the given formula we can see that the given sequence is a GP.

(Note: either formula can be used)

Arithmetic Sequences

Up Next

Adding the terms of a sequence

Return to Module 1: Sequences and Series