Geometric sequences
What is a sequence?
A sequence is the arrangement of a set of objects or numbers in order, followed by some rule. If $$a_{1}, a_{2}, a_{3}, a_{4}, \cdots$$ denote the terms of a sequence, $$1, 2, 3, 4, \cdots $$ denote the position of the terms. A sequence can be defined as either a finite sequence or an infinite sequence based on the number of terms.
Some of the most common sequences are:
Arithmetic Sequence
Geometric Sequence
Harmonic Sequence
In this article, you will learn about the geometric sequence.
A sequence in which every term is obtained by multiplying or dividing a particular number with the preceding number is a geometric sequence.
For example, the geometric progression, $$5, 10, 15, 20, 25, \cdots$$, has a common ratio of $$5$$.
E2.7A: Continue a given number sequence
Divide each element by the element before it and compare the quotient. If it is the same, there is a common ratio. And the sequence is known as the geometric sequence.
Worked example of continuing a sequence
Example 1: Are the sequences geometric?
$$1, 2, 4, 8, 16, \cdots$$
$$48, 12, 4, 2, \cdots$$
Step 1: Write the given sequences.
$$1, 2, 4, 8, 16, \cdots$$
$$48, 12, 4, 2, \cdots$$
Step 2: Divide each term by the previous term.
$$\frac{2}{1}=2$$
$$\frac{4}{2}=2$$
$$\frac{8}{4}=2$$
$$\frac{16}{8}=2$$
And
$$\frac{12}{48}=\frac{1}{4}$$
$$\frac{4}{12}=\frac{1}{3}$$
$$\frac{2}{4}=\frac{1}{2}$$
Step 3: Recall the geometric sequence rule.
Divide each term by the previous term and compare the quotients. If they are the same, a common ratio exists; and the sequence is geometric.
Step 4: Determine whether the sequences are geometric.
The first sequence is geometric because there is a common ratio. The other sequence is not geometric; it does not have a common ratio.
E2.7B: Recognise patterns in a sequence, including the term-to-term rule and relationships between different sequences.
A geometric sequence is a sequence in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence.
Each element of a geometric sequence increases or decreases by a constant factor called the common ratio.
Worked example of recognising sequences
Example 1: Is the given sequence geometric? If so, find the common ratio.
$$1, 2, 4, 8, 16, \cdots$$
Step 1: Write the given sequence.
$$1, 2, 4, 8, 16, \cdots$$
Step 2: Divide each item by the previous term.
$$\frac{2}{1}=2$$
$$\frac{4}{2}=2$$
$$\frac{8}{4}=2$$
$$\frac{16}{8}=2$$
Step 3: Recall the rule of the geometric sequence.
Divide each term by the previous term and compare the quotients. If they are the same, the order is geometric.
Step 4: Determine whether the sequence is geometric.
The given sequence is geometric because the elements have a common ratio.
Step 5: Recall the rule of the common ratio.
A geometric sequence is a sequence in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence.
Step 6: Write the common ratio.
The constant is $$2$$.
Therefore, the common ratio of the given sequence is $$2$$.
E2.7C: Find and use the $$n^{\text{th}}$$ term of sequence.
If $$a_{1}$$ is the initial term of a geometric sequence and $$r$$ is the common ratio, the sequence will be $${a_{1}, a_{1}r, a_{1}r^{2}, \cdots}$$.
The formula for the geometric sequence is $$x_{n}=a_{1}r^{n-1}$$, where $$x_{n}$$ is the $$n^{\text{th}}$$ term.
Worked example of using the $$n^{\text{th}}$$ term
Example 1: If $$10, 30, 90, 270, 810, \cdots$$ is a sequence, find the common ratio, the $$n^{\text{th}}$$ term and the $$10\text{th}$$ term.
Step 1: Write the given sequence.
$$10, 30, 90, 270, 810, \cdots$$
Step 2: Recall the rule of the common ratio.
A geometric sequence is a sequence in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence.
Step 3: Divide each term by the previous term.
$$\frac{30}{10}=3$$
$$\frac{90}{30}=3$$
$$\frac{270}{90}=3$$
$$\frac{810}{270}=3$$
Step 4: Write the common ratio.
The constant is $$3$$.
Therefore, the common ratio is $$r=3$$
Step 5: Recall the formula for the geometric sequence.
If $$a_{1}$$ is the initial term of a geometric sequence and $$r$$ is the common ratio, the sequence will be $${a_{1}, a_{1}r, a_{1}r^{2}, \cdots}$$.
The formula for the geometric sequence is $$x_{n}=a_{1}r^{n-1}$$, where $$x_{n}$$ is the $$n^{\text{th}}$$ term.
Step 6: Substitute the values in the formula.
$$x_{n}=10\times {3}^{n-1}$$
The $$n\text{ th }$$ term is $$x_{n}=10\times {3}^{n-1}$$
Step 7: Substitute $$n=10$$ in $$x_{n}=10\times {3}^{n-1}$$
$$x_{10}=10\times {3}^{10-1}$$
$$10\times {3}^{9}$$
$$10\times (19683) =196830$$
Therefore, the $$10\text{th}$$ term is $$196830$$.