# 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 $$196830$$.