Arithmetic Sequences

The interest portion of monthly payments made to pay off an automobile or home loan, and the list of maximum daily temperatures in one area for a month, are all sequences.

Create learning materials about Arithmetic Sequences with our free learning app!

• Flashcards, notes, mock-exams and more
• Everything you need to ace your exams

A sequence is a set of elements that are placed consecutively. In mathematics, we have the arithmetic sequence, geometric sequence, quadratic sequence, and others.

Arithmetic sequence definition

An arithmetic sequence is a set of ordered numbers that have a common difference between every two consecutive terms.

Each number in the set of a sequence is called a term.

An arithmetic sequence is also known as an arithmetic progression.

We take the following arithmetic sequence $2,4,6,8,10...$. Notice that when adding 2 to any term, we will get the next term. So, the common difference between every two successive terms is 2.

Another example is $5,8,11,14,...$ The difference between every two consecutive terms in this sequence is 3.

Arithmetic sequence terms

First Term

The first term of an arithmetic sequence is the first element in the sequence.

In the sequence $5,8,11,14,17$, the first term is 5 because it is the first number of the sequence. The first term is usually denoted by $a$.

Common Difference

The common difference of a sequence is the difference between two consecutive terms of the arithmetic sequence.

It is a constant and it is gotten by subtracting two consecutive terms.

Adding the common difference to one of the terms will give you the next term. The common difference is usually denoted by $d$.

Not every sequence is an arithmetic sequence. You have to first identify the type of sequence you have before attempting to solve it. And the way to do this is by knowing a common difference. If the common difference is constant throughout the sequence, then it is an arithmetic sequence.

For the sequence $2,4,6,8,...$ the common difference is 2 throughout because

$8-6=2,6-4=2,4-2=2$.

This is an arithmetic sequence.

For the sequence $,-1,2,4,8,10,...$ the common difference is not constant because

$10-8=2,8-4=4,4-2=2,2-\left(-1\right)=2+1=3$

This is not an arithmetic sequence.

The nth term

The nth term in an arithmetic sequence refers to any term in the sequence.

It is given by

${a}_{n}=a+\left(n-1\right)d$

where $n$ denotes the index of the term.

To calculate ${a}_{1}$, we substitute n by 1 in the expression of ${a}_{n}$, same goes for the other terms of the sequence.

Another form of expressing an arithmetic sequence is,

$a,a+d,a+2d,a+3d,....$

where $a$ is the first term

$d$ is the common difference.

This means that if we know the first term of an arithmetic sequence and we know what the common difference should be, then we would be able to get all the terms of the sequence.

Having the first term of an arithmetic sequence as 6 and the common difference as 3, then the sequence will be

$6,9,12,15,...$

Arithmetic sequence formula

The arithmetic sequence formula is given by the formula for the nth term of an arithmetic sequence. The formula is below.

${a}_{n}=a+\left(n-1\right)d$ where ${a}_{n}$ is the nth term,

$a$is the first term,

$n$ is the position of the term,

$d$ is the common difference

This formula is the general formula used in finding the terms of an arithmetic sequence.

Arithmetic sequence examples

Find the next three terms of the arithmetic sequence below.

$4,7,10,13,16,...$

Solution

First, we identify the first term which is 4.

Next, we find the common difference by subtracting two consecutive terms from the sequence.

$7-4=3,10-7=3$

So, the common difference is 3.

Now that we know the first term and the common difference, we can find the next three terms of the sequence by adding the common difference to the last term in the sequence.

The last known term in the sequence is 16. So, we will add 3 to 16, to get

$16+3=19$

We are to look for the next three terms, so we would do this two more times. The last term is no longer 16 but 19. So, we would add 3 to 19 to get,

$19+3=22$

We will now add 3 to 22 to get,

$22+3=25$

So, the next three terms are 19, 22 and 25.

The arithmetic sequence will be $4,7,10,13,16,19,22,25$,....

Let's take another example.

Find the next three terms of the arithmetic sequence

$5,3,1,-1,-3,...$

Solution

To solve this, we must know the first term and the common difference. The first term is 5 and the common difference is -2.

We get the common difference by subtracting two consecutive numbers in the sequence to get the difference,

$3-5=-2,1-3=-2$.

Therefore,

$d=-2$.

To get the next three terms, we will add the common difference to the last term.

The last known term is -3. So, the next term will be,

$\left(-3\right)+\left(-2\right)=-5$.

We are to get three terms, so we will repeat this step twice again.

The last term is now -5. So, the next term will be,

$\left(-5\right)+\left(-2\right)=-7$.

The next term will be,

$\left(-7\right)+\left(-2\right)=-9$.

The next three terms of the arithmetic sequence are thus $-5,-7,-9$.

Therefore, the arithmetic sequence is: $5,3,1,-1,-3,-5,-7,-9$,....

We have looked at examples where you are asked to find consequent terms of an arithmetic sequence. There will be situations where you will be asked to find a particular term in a sequence.

You can be asked to find the 9th term or 5th term or 100th term of a sequence.

To do this, you will need to use the formula below,

${a}_{n}=a+\left(n-1\right)d$

Let’s take some examples.

Find the third term of the arithmetic sequence below,

$7,14,__,28,35$.

Solution

We are asked to find the third term of the sequence. We will use the formula below.

${a}_{n}=a+\left(n-1\right)d$

Let us define all the parameters of the formula,

${a}_{n}=?\phantom{\rule{0ex}{0ex}}a=7\left(thefirsttermofthesequence\right)\phantom{\rule{0ex}{0ex}}n=3\left(becausewearelookingforthethirdterm\right)\phantom{\rule{0ex}{0ex}}d=7\left(because14-7=7,35-28=7\right)$

Now, let's substitute into the formula,

${a}_{3}={a}_{1}+\left(3-1\right)d=7+\left(3-1\right)7=7+\left(2\right)7=7+14=21$

Therefore, the third term is 21.

Find the missing terms in the arithmetic sequence below.

.$__,__,-20,-12,-4$

Solution

We are asked to find the missing terms in the sequence and usually, we use the first term in the sequence to do this. In this case, the first term is one of the missing terms. So how would we go about this?

We will first find the common difference and think of a way to use it to find the missing terms.

To get the common difference, we subtract two consecutive terms.

$d=\left(-4\right)-\left(-12\right)=8$.

We can’t use the formula we used in the previous examples because of the absence of the first term. So, let’s look for a pattern we can follow. Let’s label the first term as $x$ and the second as $y$.

So the sequence will be

$x,y,-20,-12,-4$

The relationship between each term is the common difference. If you find the difference between any two consecutive terms, you will get the same number.

Thus, if

$\left(-4\right)-\left(-12\right)=8;\left(-12\right)-\left(-20\right)=8$.Then,

$\left(-20\right)-y=8;y-x=8$

We take the first equation,

$\left(-20\right)-y=8$

We will solve for y by collecting like terms and making it the subject of the formula.

$y=-20-8=-28$

This means that the second term is $-28$.

Recall, $y-x=8$ We now know the value of $y$ as $-28$, thus

$-28-x=8\phantom{\rule{0ex}{0ex}}-x=8+28=36\phantom{\rule{0ex}{0ex}}x=-36\phantom{\rule{0ex}{0ex}}$

This means the first term is $-36$.

So, the sequence is $-36,-28,-20,-12,-4$.

Let's take another example.

Find the 17th term of the arithmetic sequence below,

$2,8,14,20,26,...$.

Solution

To find the 17th term of the sequence, we will have to use the formula below.

${a}_{n}=a+\left(n-1\right)d$

${a}_{n}=?\phantom{\rule{0ex}{0ex}}a=2\left(thefirstterm\right)\phantom{\rule{0ex}{0ex}}n=17\left(becausewearelookingforthe17thterm\right)$

Let's find the common difference d.

$d=26-20=6$

Now we will find the 17th term,

${a}_{17}=2+\left(17-1\right)6=2+\left(16\right)6=2+96=98$

The 17th term is 98.

Arithmetic sequence sum formula

Let ${\left({a}_{n}\right)}_{n}$ be an arithmetic sequence denoted by ${a}_{1},{a}_{2},{a}_{3},.......,{a}_{n},...$.

The sum of the first n terms will be the addition of each term as follows,

${S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+.....+{a}_{n}$.

If the number of terms in the sequence is so much that you can't add them or you do not know the value of all the terms, you will need a formula to be able to get the sum.

There are two formulas used in solving the sum of an arithmetic mean. The first formula is given by,

${S}_{n}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

where ${S}_{n}$is the sum of the arithmetic sequence,

$n$ is the number of terms in the sequence,

$a$ is the first term,

$d$ is the common difference.

This formula is used when the last term of the sequence is not known.

The other formula is given by,

${S}_{n}=\frac{n}{2}\left[a+{a}_{n}\right]$

where ${S}_{n}$is the sum of the sequence,

$n$ is the number of terms in the sequence,

$a$ is the first term,

${a}_{n}$ is the last term,

This formula is used when the last term is known. Let’s take some examples.

Find the sum of the arithmetic sequence $5,9,13,17,...$ up to 10 terms.

Solution

From the question, we are given the first term which is 5 but we do not know the last term. But we know that the number of terms is 10. The formula to use when the last term is not known is.

${S}_{n}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

$a=5,n=10$

The common difference is,

$d=9-5=4$

Now, we will substitute the values in the formula, ${S}_{n}=\frac{10}{2}\left[2\left(5\right)+\left(10-1\right)4\right]=5\left[10+\left(9\right)4\right]=5\left[10+36\right]=5\left[46\right]=230$

The sum of the terms is 230.

Let's take another example.

Find the sum of an arithmetic progression of 4 terms whose first term is 4 and its last term is 19.

Solution

Here, we know what the first and the last term are. The formula to use in this case is,

${S}_{n}=\frac{n}{2}\left[a+{a}_{n}\right]$

$n=4,a=4,{a}_{4}=19$

We will now substitute the values in the formula,

${S}_{n}=\frac{4}{2}\left[4+19\right]=2\left[23\right]=46$

The sum of the 4 terms is 46.

Arithmetic Sequences - Key takeaways

• An arithmetic sequence is a set of numbers where the difference between each term is the same, that is, it is constant.
• The common difference of an arithmetic sequence is the difference between the terms of the arithmetic sequence. It is a constant and it is gotten by subtracting two consecutive terms. Adding the common difference to one of the terms will give us the next term. The common difference is denoted by $d$.
• The general formula used in finding the terms of an arithmetic sequence is ${a}_{n}={a}_{1}+\left(n-1\right)d$.
• To find the sum of the first n terms of an arithmetic sequence, we use one of the formulae, ${S}_{n}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$ or ${S}_{n}=\frac{n}{2}\left[a+{a}_{n}\right]$.

Flashcards in Arithmetic Sequences 4

Learn with 4 Arithmetic Sequences flashcards in the free StudySmarter app

We have 14,000 flashcards about Dynamic Landscapes.

What is the term arithmetic sequence?

An arithmetic sequence is a set of numbers where the difference between each term is the same, that is, it is constant.

How do you find the nth term in an arithmetic sequence?

To find the nth term of an arithmetic sequence, the formula is below is used.

Where  is the nth term

is the first term

is the position of the term

is the common difference

What is the formula for arithmetic sequence?

The formula used to find the terms of an arithmetic sequence is  a_n=a+(n-1)d . The subsequent terms of an arithmetic sequence can also be found by adding the common difference to the last known term.

What is an arithmetic sequence example?

An arithmetic sequence example is 2, 4, 6, 8, 10, ..... . It is an arithmetic sequence because it has a consistent common difference.

How do you find the sum of an arithmetic sequence?

The sum of an arithmetic sequence can be found using two formulae.

S= (n/2) [ 2a+ ( n - 1 ) d ]

The other formula is used when the last term is known.

S= (n/2) [ a + a_n ]

StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

StudySmarter Editorial Team

Team Math Teachers

• Checked by StudySmarter Editorial Team

Join over 22 million students in learning with our StudySmarter App

The first learning app that truly has everything you need to ace your exams in one place

• Flashcards & Quizzes
• AI Study Assistant
• Study Planner
• Mock-Exams
• Smart Note-Taking