# Arithmetic sequences

Last updated: September 29, 2021

## What is a sequence

A sequence is a set of numbers that usually follows a specific pattern in mathematics. Each element in the sequence is called an item. Some of the most common examples of sequences are:

• Arithmetic Sequence

• Geometric Sequence

• Harmonic Sequence

The order of generating each element by adding or subtracting a number from the previous number is an arithmetic sequence.

## E2.7A: Continue a given number sequence

A number sequence is a sequence or an ordered list controlled by a pattern or rule. Number in a sequence are called terms. A sequence that never ends is an infinite sequence. At the same time, a sequence with an end is known as a finite sequence.

### Worked example of a sequence

Example 1: Which list of numbers makes a sequence?

$$4$$, $$6$$, $$3$$, $$10$$, $$14$$, $$15$$, $$\cdots$$

$$1$$, $$4$$, $$7$$, $$10$$, $$13$$, $$\cdots$$

Step 1: State the relevant information provided

$$4$$, $$6$$, $$3$$, $$10$$, $$14$$, $$15$$, $$\cdots$$

$$1$$, $$4$$, $$7$$, $$10$$, $$13$$, $$\cdots$$

Step 2: Write the list of numbers that makes a sequence.

The first list of numbers does not form a sequence because the arrangement of the numbers lacks order or pattern.

The other list is a sequence because there is a fixed way of obtaining the preceding number. The consecutive number is obtained by adding $$3$$ to the preceding integer.

## E2.7B: Recognise patterns in a sequence, including the term-to-term rule and relationships between different sequences

An arithmetic sequence is a list of numbers with a specific pattern. If there is a sequence and you took a number and subtract it from the previous number, the result will always be the same; because it is an arithmetic sequence.

The constant difference of all consecutive pairs in the sequence is called the common difference.

• If the common difference is positive, then the sequence is increasing.

• If the common difference is negative, then the sequence is decreasing.

The common difference of an arithmetic sequence is $$d=a_{n}-a_{n-1}$$, where $$a_{n}$$ is the last term in the sequence, and $$a_{n-1}$$ is the previous term in the sequence.

### Worked example of recognising a sequence

Example 1: What is the common difference in the following order?

$$-4$$, $$-2$$, $$0$$, $$2$$, $$4$$, $$6$$, $$\cdots$$

Step 1: Recall the formula for the common difference.

The common difference of an arithmetic sequence is $$d=a_{n}-a_{n-1}$$, where $$a_{n}$$ is the last term, and $$a_{n-1}$$ is the previous term in the sequence.

Step 2: Write the difference of the given sequence.

$$6-4=2$$

$$4-2=2$$

$$2-0=2$$

$$0-(-2)=2$$

$$(-2)-(-4)=2$$

Each number is $$2$$ numbers away from the previous number.

Therefore, the common difference in the given sequence is $$2$$.

## E2.7C: Find and use the $$n^{\text{th}}$$ term of a sequence.

The formula of the arithmetic sequence allows you to directly calculate each member of an arithmetic sequence. In mathematical words, the formula of an arithmetic sequence is designated to the $$n^{\text{th}}$$ term of the sequence. For instance, in $$1$$, $$5$$, $$9$$, $$13$$, $$17$$, there is a constant difference between the terms.

Arithmetic sequence formula: $$a_{n}=a_{1}+(n-1)d$$

Here, $$a_{n}$$ is the $$n^{\text{th}}$$ term in the sequence, $$a_{1}$$ is the first term in the sequence, $$n$$ is the term number, and  $$d$$ is the common difference.

### Worked example of using $$n^{\text{th}}$$ term

Example 1: If $$2$$, $$12$$, $$22$$, $$32$$, $$42\cdots$$ is a sequence, find the common difference, the $$n^{\text{th}}$$ term and the $$21{\text{st}}$$ term.

Step 1: Write the given sequence.

$$2$$, $$12$$, $$22$$, $$32$$, $$42$$, $$\cdots$$

Step 2: Recall the formula for the common difference.

The common difference of an arithmetic sequence is $$d=a_{n}-a_{n-1}$$, where $$a_{n}$$ is the last term, and $$a_{n-1}$$ is the previous term in the sequence.

Step 3: Write the difference of the given sequence.

$$42-32=10$$

$$32-22=10$$

$$22-12=10$$

$$12-02=10$$

Step 4: Write the common difference.

Each number is $$10$$ numbers away from the previous number.

Therefore, the common difference of the given sequence is, $$d=10$$

Step 5: Write the formula for the arithmetic sequence.

Arithmetic sequence formula: $$a_{n}=a_{1}+(n-1)d$$

Step 6: Substitute the values in the above formula.

$$a_{n}=2+(n-1)10$$

$$2+10n-10$$

$$10n-8$$

So, the $$n^{\text{th}}$$ of the given sequence is $$a_{n}=10n-8$$

Step 7: Draw the graph of the sequence.

The graph of the sequence $$a_{n}=10n-8$$ shows a slope of $$10$$ and a vertical intercept of $$-8$$.

Step 8: Substitute $$n=21$$ in $$a_{n}=10n-8$$

$$a_{21}=10(21)-8$$

$$210-8=202$$

So, the $$21\text{st}$$ term is $$202$$.