Homebreadcumb forward arrow iconBlogbreadcumb forward arrow iconArtificial Intelligencebreadcumb forward arrow iconArithmetic Progression Formula: Everything You Need to Know

Arithmetic Progression Formula: Everything You Need to Know

Last updated:
9th Feb, 2021
Read Time
5 Mins
share image icon
In this article
Chevron in toc
View All
Arithmetic Progression Formula: Everything You Need to Know


An arithmetic progression is a sequence in which the next term in the sequence is obtained by adding a constant to each term. The constant added is called the common difference. It is a sequence such that the difference between any two consecutive terms in the sequence is always a constant.

Suppose, n1, n2, n3……..nn are the

terms of an arithmetic progression sequence.

Then, n2 = n1 + d, n3 = n2 + d and so on.

Ads of upGrad blog

Where n1 = the first term and d is the common difference

Best Machine Learning and AI Courses Online

Arithmetic Progression Examples

Verify if the following sequence 3, 6, 9, 12, 15 is an arithmetic progression or not.
For this sequence to be an arithmetic progression sequence, the common difference between the consecutive terms should be constant.

Common difference (d) = n2 – n1 must equal to n3 – n2 and so on.

In this sequence, d = 6 – 3 = 3, 9 – 6 = 3, 12 – 9 = 3, and 15 – 12 = 3.

The difference between consecutive terms is constant. Hence, the above sequence is an arithmetic progression.

Also Read: Solve Problems Using RNN

Arithmetic Progression Formula

To understand the arithmetic progression formula, one should be familiar with the terminologies used in the formula.


As the name states, the first term is the first term of the sequence, which is usually represented by n1. For example, in the 5, 12, 19, 26, 33 sequence, the first term is 5.

In-demand Machine Learning Skills

Common Difference

A common difference is the fixed number that is added or subtracted between two consecutive terms (except the first term) in the arithmetic progression. It is denoted by ‘d’.

For example, if n1 is the first term, then:

n2 = n1 + d

n3 = n2 + d and so on

Arithmetic Progression Formula to Find the General Term or nth Term

The general term or nth term in an arithmetic progression is found by:

Nn = a + (n-1) *d

where ‘a’ is the first term and ‘d’ is a common difference.

So, 1st term, N1 = a + (1-1) *d

2nd term, N2 = a + (2-1) *d

3rd term, N3 = a + (3-1) *d

By computing ‘n’ terms in the above formula, we get the general form of an arithmetic progression.

a, a + d, a + 2d, a + 3d, …… a + (n-1) *d

Arithmetic Progression Formula to Find the Sum

The arithmetic progression formula for the sum of ‘n’ terms where ‘a’ is the first term and ‘d’ is a common difference is as follows.

When the nth term is unknown:

Sn = (n/2) * [2a + (n − 1) * d]

When the nth term is known:

Sn = (n/2) * [a1 + an]

Formula derivation

Let us assume that ‘t’ is the nth term of the series and Sn is the sum of first n terms in an arithmetic progression:  a, (a + d), (a + 2d), …., a + (n – 1) * d.


Sn = a1 + a2 + a3 + ….an-1 + an

Substituting the terms in the above formula, we get

Sn = a + (a + d) + (a + 2d) + …….. + (t – 2d) + (t – d) + t                  …(1)

After writing the equation (1) in the reverse order

Sn =t + (t – d) + (t – 2d) + …….. + (a + 2d) + (a + d) + a                  …(2)

Now, add equation (1) and (2), we get

2Sn = (a + t) + (a + t) + (a + t) + …….. + (a + t) + (a + t) + (a + t)

2Sn = n * (a + t)

Sn = (n/2) * (a + t)                                                                               …(3)

Let us replace the last term ‘t’ by the nth term in equation 3, we get,

nth term = a + (n – 1) * d

Sn = (n/2) * {a + a + (n – 1) * d}

Sn = (n/2) * {2a + (n – 1) * d}


If you are asked to find the sum of the first 30 terms of a sequence 5, 11, 17, 23, ……


a = 5, d = a2 – a1 = 11 – 5 = 6

Sn = (n/2) * {2a + (n – 1) * d}

Sn = (30/2) * (2 * 5 + (35 – 1) * 6}

Sn = (15) * (10 + 204)

Sn = 15 * 214

Sn = 3210

Ads of upGrad blog

Popular AI and ML Blogs & Free Courses


In mathematics, an arithmetic progression is a series of numbers where the difference between two consecutive terms is always constant. We can find multiple examples of arithmetic progression in our daily life. For example, enrollment numbers of students in a batch, months in a year, etc. 

Today, we stand on the cusp of a medical revolution, all thanks to machine learning and artificial intelligence. However, using technology alone will not improve healthcare. There also needs to be curious and dedicated minds who can give meaning to such brilliant technological innovations as machine learning and AI.

Learn ML Course from the World’s top Universities. Earn Masters, Executive PGP, or Advanced Certificate Programs to fast-track your career.


Pavan Vadapalli

Blog Author
Director of Engineering @ upGrad. Motivated to leverage technology to solve problems. Seasoned leader for startups and fast moving orgs. Working on solving problems of scale and long term technology strategy.
Get Free Consultation

Selectcaret down icon
Select Area of interestcaret down icon
Select Work Experiencecaret down icon
By clicking 'Submit' you Agree to  
UpGrad's Terms & Conditions

Our Popular Machine Learning Course

Frequently Asked Questions (FAQs)

1What are the different types of progressions in mathematics?

Numbers are sorted in a predictable order when they are arranged in a progression. Progressions have the ability to anticipate the next numbers in a series in a given set of integers. There are three different types of progressions that are used in mathematics, namely, arithmetic progression (AP), harmonic progression (HP) and geometric progression (GP). In AP, the common difference is used to find the next term, in GP, common ratio is used while HP basically means that the reciprocal of the given terms are in AP.

2What are the two types of arithmetic progression series?

There are two types of arithmetic progression series in mathematics- finite series and infinite series. In finite series, the number of terms are either known or at least it is given that they are limited. While in an infinite sequence, the number of terms are infinite. For finding the common difference, the formula is the same for both the arithmetic progression series. But when it comes to finding the sum, the formula differs.

3How is an arithmetic progression related to harmonic progression?

In an arithmetic progression, the common difference is taken out, and then, using the first term and the common difference, the sum of the series is calculated. When it comes to harmonic progression, there isn’t any difference between finding the common difference and the sum of the series. The terms of the given HP are reciprocated, and then the same formula as AP is used. Thus, when the terms of the HP are reciprocated, the series becomes an AP. That is how AP and HP are connected.

Explore Free Courses

Suggested Blogs

Top 5 Natural Language Processing (NLP) Projects & Topics For Beginners [2024]
What are Natural Language Processing Projects? NLP project ideas advanced encompass various applications and research areas that leverage computation
Read More

by Pavan Vadapalli

30 May 2024

Top 8 Exciting AWS Projects & Ideas For Beginners [2024]
AWS Projects & Topics Looking for AWS project ideas? Then you’ve come to the right place because, in this article, we’ve shared multiple AWS proj
Read More

by Pavan Vadapalli

30 May 2024

Bagging vs Boosting in Machine Learning: Difference Between Bagging and Boosting
Owing to the proliferation of Machine learning applications and an increase in computing power, data scientists have inherently implemented algorithms
Read More

by Pavan Vadapalli

25 May 2024

45+ Best Machine Learning Project Ideas For Beginners [2024]
Summary: In this Article, you will learn Stock Prices Predictor Sports Predictor Develop A Sentiment Analyzer Enhance Healthcare Prepare ML Algorith
Read More

by Jaideep Khare

21 May 2024

Top 9 Python Libraries for Machine Learning in 2024
Machine learning is the most algorithm-intense field in computer science. Gone are those days when people had to code all algorithms for machine learn
Read More

by upGrad

19 May 2024

Top 15 IoT Interview Questions & Answers 2024 – For Beginners & Experienced
These days, the minute you indulge in any technology-oriented discussion, interview questions on cloud computing come up in some form or the other. Th
Read More

by Kechit Goyal

19 May 2024

40 Best IoT Project Ideas & Topics For Beginners 2024 [Latest]
In this article, you will learn the 40Exciting IoT Project Ideas & Topics. Take a glimpse at the project ideas listed below. Best Simple IoT Proje
Read More

by Kechit Goyal

19 May 2024

Top 22 Artificial Intelligence Project Ideas & Topics for Beginners [2024]
In this article, you will learn the 22 AI project ideas & Topics. Take a glimpse below. Best AI Project Ideas & Topics Predict Housing Price
Read More

by Pavan Vadapalli

18 May 2024

Image Segmentation Techniques [Step By Step Implementation]
What do you see first when you look at your selfie? Your face, right? You can spot your face because your brain is capable of identifying your face an
Read More

by Pavan Vadapalli

16 May 2024

Schedule 1:1 free counsellingTalk to Career Expert
footer sticky close icon