# Arithmetic sequences

**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

In this article, you will learn about the arithmetic 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 $$202$$.