BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon103 Learners

Last updated on July 4th, 2025

Math Whiteboard Illustration

Arithmetic Progression

Professor Greenline Explaining Math Concepts

In mathematics, a sequence is a set or list of numbers that are arranged in a particular order. An arithmetic progression (AP) is a sequence where the terms are arranged in such a way that the difference between any two successive terms is constant. In this article, we will learn more about arithmetic progression with examples.

Arithmetic Progression for Saudi Students
Professor Greenline from BrightChamps

What is an Arithmetic Progression?

A sequence where a term is found by adding a constant to its previous term is called an arithmetic progression. In other words, the difference between two consecutive terms in an AP will be the same. The difference is known as the common difference (d), and the first term is denoted by a or a1, and the nth term is denoted by an. For example, 3, 6, 9, 12, 15, … is an arithmetic progression. Here, a is 3, and the constant d is also 3. In its general form, AP can be expressed as a, a + d, a + 2d,…, a + (n - 1)d.

Professor Greenline from BrightChamps

Difference Between AP and GP

There are mainly two types of progressions: arithmetic and geometric (GP). Here, we will be discussing the difference between AP and GP:

 

Arithmetic Progression Geometric Progression
  • AP is formed by adding a fixed number to its previous term
  • GP is a sequence formed by multiplying a constant term by the previous term.
  • Here, the difference between two consecutive terms will be the same, and it is known as the common difference (d)
  • In GP, the ratio of every term will be the same. The ratio is known as the common ratio (r)
  • The general form of AP: a, a + d, a + 2d, a + 3d, …, a + (n - 1)d
  • The general form of GP: a, ar, ar2, ….., arn - 1
  • The nth term formula of AP: 
    an = a + (n - 1)d
  • The formula for the nth in GP: an = arn-1
  • The sum of the n terms in AP: 
    Sn = (n/2)[2a + (n - 1)d]
  • Sum of the n terms in GP: 
    Sn = a(rn - 1)(r - 1)
Example: 5, 10, 15, 20, 25, … Example: 3, 6, 12, 24, ….

 

Professor Greenline from BrightChamps

What is the Formula for AP?

Key formulas for arithmetic progression for calculating common difference, the nth term, and the sum of terms. 

The common difference in AP: 
The common difference of a sequence is the difference between two consecutive terms, and it is denoted by d:
d = a2 - a = a3 - a2 = a4 - a3 = … = an - an-1

The nth term of an AP: 
The nth term of an AP can be expressed as an = a + (n - 1)d

Where n is the number of terms
a is the first term 
a2 is the second term
a3 is the third term
d is the common difference
an is the nth term of the sequence

Now we will learn how to find the sum of the first n terms of an AP. So let’s consider the first n terms of an AP as a1, a1 + d, a1 + 2d, a1 + 3d, ….., a1 + (n - 1)d, and the sum of these n terms is sn
Then the sum of nth terms:
Sn = a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) + ….., + (a1 + (n - 1)d) 

Now let's reverse the sequence and add corresponding terms. It can be written as:
Sn = an + (an - d) + (an - 2d) + (an - 3d) + ….., + (an + (n - 1)d) 

Adding these two equations: 
Sn = a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) + ….., + (a1 + (n - 1)d) 
Sn = an + (an - d) + (an - 2d) + (an - 3d) + ….., + (an + (n - 1)d) 

By adding these, we get: 2sn = (a1 + an) + (a1 + an) + (a1 + an) + …. + (a1 + an)
All the d terms are cancelled out here, so:
2sn = n(a1 + an)
Sn = n(a1 + an)/2, let's consider it as equation 1

Substituting the formula to find an in equation 1, an = a1 + (n - 1)d
Then, Sn = n/2 (a1 + a1 + (n - 1)d) 
= n/2 [(2a + (n - 1)d], let’s consider this equation as 2

Equations 1 and 2 are used to find the sum of an arithmetic progression.  
So, Sum of nth term: sn = n/2 (2a + (n - 1)d) and sn = n/2(a + l)

Professor Greenline from BrightChamps

Nth Term of Arithmetic Progression

The nth term of an AP is used to find any term in the sequence without listing all the previous terms. There is a formula that we can use to find the nth term, which is an = a + (n - 1)d. 
For example, for the AP: 2, 4, 6, 8, 10,… we find the nth term. Here, a = 2, d = 2
an = a + (n - 1)d
= 2 + (n - 1)2
= 2 + 2n -2
an = 2n

Professor Greenline from BrightChamps

What are the Uses of AP Formula for General Term?

To find the nth term of an AP, we use the formula an = a + (n - 1)d. For example, for the sequence 10, 20, 30, 40, …, here, a = 10 and d = 10. 
So, the 5th term a5 = 10 + (5 - 1)10 = 10 + 40 = 50. Similarly, to find the 55th term, we can use the formula, 
an = a + (n - 1)d
a55 = 10 + (55 - 1) 10
= 10 + 54 × 10 
= 10 + 540 = 550. 
So, the 55th term of this AP is 550.

The arithmetic sequence explicit formula is the nth term of an AP, it is used to find any term of the sequence.

Professor Greenline from BrightChamps

What is the Sum of Arithmetic Progression?

Now we will learn to find the sum of arithmetic progression. The sum of the first ‘n’ terms can be calculated with the help of the formula mentioned below.

When the nth term is unknown, the sum of n is calculated using the formula:

Sn = n/2 (2a + (n - 1)d)

When the nth term is known, the sum of n is calculated using: Sn = n/2 (a + an)
 

Where,

n is the number of terms

a is the first term

d is the common difference

a is the first term

an is the nth term 

Professor Greenline from BrightChamps

Real-Life Applications of Arithmetic Progression

In our real world, we use AP from basic counting to calculate the interest rate. Here, we will learn some applications of arithmetic progression.

 

  • We use AP to predict the upcoming term in a sequence. This is achieved by adding a constant difference. For e.g., we can predict the next number in the sequence 3, 7, 11,.....

 

  • For predicting future values based on past trends, we use an AP

 

  • To calculate the interest rates, appropriations, and loan disbursements, we use AP. 

 

  • For seating arrangements such as arranging seats in rows and columns with equal distance, we use AP
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Arithmetic Progression

When working on arithmetic progression, students often make mistakes, which leads to errors. Here are some common mistakes and the ways to avoid them:

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing the common difference with the first term

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Confusing the common difference and the first term can lead to errors. So always remember that ‘a’ is the first term and d is the common difference.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Errors in finding the common difference

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

In finding the common difference, mostly students make errors in subtraction. For AP, it is mandatory to subtract the latter number from the previous number to find out the common difference. In some cases, the students also make errors in considering the sign.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Calculating the nth term without finding d

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

To find the nth term, it is necessary to find ‘d’, the common difference between two consecutive terms. It is always advisable to memorize the formula used for finding the nth term of AP series.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Using the wrong formula to find the sum

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

One of the common mistakes is mixing up formulas while trying to find the sum of terms. To avoid this, look at the given values and use the formulas accordingly. 

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing with AP and GP

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students tend to mix up AP and GP, as they may not check the pattern of the sequence. So, always remember that AP is formed by adding the common difference to the previous term, whereas in GP we multiply the previous term by the common ratio.

arrow-right
Max from BrightChamps Saying "Hey"

Solved Examples of Arithmetic Progression

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Find the 10th term of AP, where the first term is 5 and the common difference is 3

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The 10th term of the AP is 32

Explanation

The nth term of an AP is calculated by:

an = a + (n - 1)d

Here, a = 5, d = 3, and n = 10

So, a10 = 5 + (10 - 1) 3

= 5 + 9 × 3

= 5 + 27 = 32

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

Find the common differences between AP 2, 5, 8, 11,…

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The common difference is 3

Explanation

The common difference is the difference between the two consecutive terms.

d = 5 - 2 = 3

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

Find the sum of all multiples of 7 between 50 and 200?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The sum of all the multiples of 7 between 50 and 200 is 2646

Explanation

The first multiple of 7 that is greater than or equal to 50 is 56

The last multiple of 7 that is less than or equal to 200 is 196

Here, the d = 7

a = 56

an = 196

So, n = an- a/d + 1

= (196 - 56 / 7) + 1

= 20 + 1 = 21

 

Sum of first n terms: Sn = n/2 (a + an)

= 21/2 (56 + 196) 

= 21 × 126

2646

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

Find the 15th term of the AP where a = 2 and d = 15?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The 15th term is 212

Explanation

 Here, the first term (a) is 2

The common difference (d) is 15

an = a + (n - 1)d

So, a15 = 2 + (15 - 1)15

= 2 + 14 × 15 = 212

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

Find the number of terms in the AP 7, 13, 19, …, 205.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The number of terms in the AP 7, 13, 19, … , 205 is 34.

Explanation

Here a = 7, d = 6, and an = 205

an = a + (n - 1)d

205 = 7 + (n - 1) 6

205 - 7 = 6n -6

198 + 6 = 6n 

204 = 6n

n = 204 / 6 = 34

So, the number of terms is 34. 

Max from BrightChamps Praising Clear Math Explanations
Ray Thinking Deeply About Math Problems

FAQs on Arithmetic Progression

1.What is the AP and GP formula?

Math FAQ Answers Dropdown Arrow

2.What is the AP basic formula?

Math FAQ Answers Dropdown Arrow

3.What is the formula to find the sum of first n terms of an AP?

Math FAQ Answers Dropdown Arrow

4.What is d in AP?

Math FAQ Answers Dropdown Arrow

5.How to find r in a geometric sequence?

Math FAQ Answers Dropdown Arrow

6.How does learning Algebra help students in Saudi Arabia make better decisions in daily life?

Math FAQ Answers Dropdown Arrow

7.How can cultural or local activities in Saudi Arabia support learning Algebra topics such as Arithmetic Progression?

Math FAQ Answers Dropdown Arrow

8.How do technology and digital tools in Saudi Arabia support learning Algebra and Arithmetic Progression?

Math FAQ Answers Dropdown Arrow

9.Does learning Algebra support future career opportunities for students in Saudi Arabia?

Math FAQ Answers Dropdown Arrow
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom