NCERT Class 10 Arithmetic Progression

Have you ever noticed patterns in everyday life—like your pocket money increasing every month, steps of a staircase, or savings growing year by year? It makes you wonder, why do numbers follow patterns so naturally? This is exactly what NCERT Class 10 Arithmetic Progression helps you understand. An Arithmetic Progression (AP) is one of the simplest and most useful number patterns. In this blog, we’ll break down every concept step by step so that by the end, you can confidently understand and solve AP questions without confusion.

What is an Arithmetic Progression? 

An Arithmetic Progression (AP) is a sequence of numbers where each term is obtained by adding a fixed number to the previous term. This fixed number is called the common difference (d). In simple words, if numbers are increasing or decreasing regularly by the same value, they form an AP.

Let’s understand the key terms:

• First term (a): The starting number of the sequence
• Common difference (d): The number added (or subtracted) each time 

For example, in the sequence 2, 5, 8, 11,

• First term (a) = 2
• Common difference (d) = 5 − 2 = 3 

This means each number increases by 3.

Arithmetic Progressions are not just in books—they appear in real life too. Imagine your salary increasing by ₹500 every year. Or saving ₹100 more than last month. Even the steps of a ladder often decrease in size in a fixed pattern. These are all examples of AP in action.

Let’s look at different types of AP:

• Increasing AP:
  3, 7, 11, 15… (numbers increase by 4)
• Decreasing AP:
  20, 15, 10, 5… (numbers decrease by 5)
• Constant AP:
  6, 6, 6, 6… (no change, difference is 0) 

Understanding AP basics is important because it forms the foundation for solving bigger problems like finding any term or calculating sums quickly. Once you understand this pattern, the rest of the chapter becomes much easier and more logical.

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Understanding Terms in AP (a, d, n, an)

To truly understand NCERT Class 10 Arithmetic Progression, you need to be clear about the basic terms used in AP. These terms help you solve almost every question in this chapter.

• a (First Term): This is the starting number of the sequence.
• d (Common Difference): The fixed number added (or subtracted) each time.
• n (Number of Terms): Total terms in the sequence.
• an (nth Term): The value of the term at position n. 

Let’s see how to identify them easily. The first term (a) is always the very first number in the sequence—no calculation needed. To find the common difference (d), subtract any term from the next term.

For example, in 2, 5, 8, 11:

• a = 2
• d = 5 − 2 = 3 

Now, understanding the value of d is important:

• If d is positive, the AP increases (numbers go up)
• If d is negative, the AP decreases (numbers go down)
• If d is zero, all terms are the same 

Let’s look at some quick examples:

• 2, 5, 8, 11 → d = +3 (Increasing AP)
• 10, 7, 4, 1 → d = −3 (Decreasing AP)
• 3, 3, 3, 3 → d = 0 (Constant AP) 

Once you understand these terms, you can easily move ahead to formulas and problem-solving. These four elements—a, d, n, and an—are the backbone of Arithmetic Progression.

How to Check if a Sequence is an AP

Sometimes, you are given a sequence and asked whether it is an Arithmetic Progression or not. The good news? There’s a very simple rule to check this.

Rule:

A sequence is an AP if the difference between consecutive terms is always the same.

Step-by-step method:

1. Subtract the first term from the second
2. Subtract the second term from the third
3. Compare the differences 

If all differences are equal → it is an AP
If not → it is not an AP

Example of a correct AP:

Sequence: 4, 9, 14, 19

• 9 − 4 = 5
• 14 − 9 = 5
• 19 − 14 = 5 

Since the difference is constant, this is an AP.

Example of an incorrect sequence:

Sequence: 2, 4, 8, 16

• 4 − 2 = 2
• 8 − 4 = 4
• 16 − 8 = 8 

Here, the differences are not the same, so this is not an AP.

Tip Box: Common Mistakes Students Make

• Subtracting in the wrong order (always do next − previous)
• Checking only one pair instead of all
• Assuming a pattern without calculating
• Confusing multiplication patterns with AP 

If you follow the steps carefully, you can quickly and confidently identify any AP.

Types of Arithmetic Progressions

Arithmetic Progressions can be grouped into different types based on how they behave. Understanding these types helps you recognize patterns faster.

1. Finite AP

A sequence with a fixed number of terms.
Example: 2, 4, 6, 8, 10
Real-life example: Number of days in a week-based savings plan.

2. Infinite AP

A sequence that continues forever with no last term.
Example: 1, 3, 5, 7, …
Real-life example: Counting numbers.

3. Increasing AP

An AP where each term is greater than the previous one (d > 0).
Example: 5, 10, 15, 20
Real-life example: Salary increasing every year.

4. Decreasing AP

An AP where each term is smaller than the previous one (d < 0).
Example: 50, 45, 40, 35
Real-life example: Fuel decreasing during a journey.

5. Constant AP

An AP where all terms are equal (d = 0).
Example: 7, 7, 7, 7
Real-life example: Fixed monthly allowance with no change.

Understanding these types makes it easier to identify patterns in both exam questions and real-life situations.

General Form of an AP

The general form of an Arithmetic Progression (AP) is written as:
a, a + d, a + 2d, a + 3d, a + 4d, …

Here,

• a is the first term
• d is the common difference 

This form helps us understand how each term is built. Instead of writing random numbers, every term follows a clear pattern. You start with a, then keep adding d again and again.

Think of it like taking equal “steps.”

• The first term is your starting point
• Each next term is one step ahead by d 

For example, if a = 2 and d = 3, the AP becomes:
2, 5, 8, 11, 14…

Here, each number is just one more “jump” of +3 from the previous one.

This “step jump” idea makes AP very easy to visualize. Imagine climbing stairs where each step is equally spaced—that’s exactly how an AP works.

Understanding this general form is important because it leads directly to formulas like the nth term and sum, making calculations faster and easier.

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Finding the nth Term of an AP (Formula Explained)

Often, we don’t want to write all terms of an AP—we just want to find a specific term, like the 20th or 50th term. This is where the nth term formula helps.

Let’s build the logic step by step:

• First term: a₁ = a
• Second term: a₂ = a + d
• Third term: a₃ = a + 2d
• Fourth term: a₄ = a + 3d 

If you observe carefully, the pattern is:

• The second term has 1 “d”
• The third term has 2 “d”
• The fourth term has 3 “d” 

So, the nth term will have (n − 1) times d.

Final Formula:

an = a + (n − 1)d

This formula helps you find any term directly without writing the whole sequence.

What does it mean in simple words?

Start with the first term (a) and add the common difference (d) repeatedly (n − 1) times.

Real-life example:

Suppose your monthly savings start at ₹100 and increase by ₹50 every month.

• a = 100
• d = 50 

To find savings in the 6th month:
an = 100 + (6 − 1) × 50
= 100 + 250
= ₹350

So instead of calculating month by month, you can directly jump to the answer. That’s the power of the nth term formula!

Solved Examples – nth Term

Let’s solve some important types of questions step by step.

Example 1: Find the 10th term

AP: 2, 7, 12, …

• a = 2
• d = 7 − 2 = 5
• n = 10 

an = a + (n − 1)d
= 2 + (10 − 1) × 5
= 2 + 45
= 47

Example 2: Which term is equal to 95?

AP: 5, 10, 15, …

• a = 5
• d = 5
• an = 95 

95 = 5 + (n − 1) × 5
95 = 5n

n = 19

So, 95 is the 19th term.

Example 3: Check if 50 is a term

AP: 3, 7, 11, …

• a = 3
• d = 4 

50 = 3 + (n − 1) × 4
50 = 4n − 1
4n = 51
n = 12.75

Since n is not a whole number, 50 is not a term of this AP.

Example 4: Find the missing term

AP: 4, __, 12

Let the missing term be x.

• First difference: x − 4
• Second difference: 12 − x 

For AP:
x − 4 = 12 − x
2x = 16
x = 8

These examples cover all common question types. Once you practice these, solving AP problems becomes quick and easy.

H2: Practice Questions – nth Term (Category Wise)

Level 1: Basic (Find nth term, identify a and d)

1. Find the 12th term of the AP: 3, 6, 9, 12…
2. Identify a and d in the AP: 7, 10, 13, 16…
3. Find the 15th term of the AP: 5, 9, 13…
4. Identify a and d in: 20, 15, 10, 5…
5. Find the 8th term of the AP: 1, 4, 7, 10…
6. Identify a and d in: 2, 2, 2, 2…
7. Find the 20th term of the AP: 4, 8, 12…
8. Identify a and d in: -3, -1, 1, 3…
9. Find the 25th term of the AP: 10, 13, 16…
10. Identify a and d in: 100, 90, 80, 70… 

Level 2: Medium (Find which term, missing values)

1. Which term of the AP: 4, 9, 14… is 99?
2. Find the term number for 145 in the AP: 5, 10, 15…
3. Which term of the AP: 3, 7, 11… is 59?
4. Find the missing term: 6, __, 14
5. Find the missing term: 10, __, 20
6. Which term of the AP: 2, 6, 10… is 82?
7. Find the term number of 200 in the AP: 8, 12, 16…
8. Find the missing term: 3, __, 9
9. Which term of the AP: 1, 4, 7… is 100?
10. Find the missing term: 15, __, 25 

Level 3: Advanced (Word problems, reverse logic)

1. A salary starts at ₹2000 and increases by ₹200 every year. Find the salary in the 15th year.
2. A student saves ₹50 in the first month and increases savings by ₹20 every month. Find savings in the 12th month.
3. The 5th term of an AP is 18 and the common difference is 3. Find the first term.
4. The 10th term of an AP is 50 and d = 4. Find the first term.
5. The 3rd term is 7 and the 8th term is 22. Find the AP.
6. A ladder has steps decreasing by 2 cm. If the first step is 50 cm, find the length of the 10th step.
7. The 7th term of an AP is 35 and the first term is 5. Find the common difference.
8. Find the term number whose value is zero in the AP: 10, 7, 4…
9. If the nth term of an AP is given as 2n + 3, find the 20th term.
10. The 4th term is 10 and the 9th term is 25. Find the first term and common difference. 

These practice questions cover all difficulty levels—from basic understanding to exam-level application—helping students build strong confidence in Arithmetic Progressions.

Sum of First n Terms of an AP (Concept)

Imagine adding numbers like this:
2 + 5 + 8 + 11 + 14 + … up to 50 terms. Sounds tiring, right? Writing and adding so many numbers takes time and increases the chance of mistakes. So, mathematicians looked for a smarter way—a shortcut.

There’s a famous story about a young mathematician, Carl Friedrich Gauss. When he was just a child, his teacher asked the class to add numbers from 1 to 100. While others were still calculating, Gauss quickly gave the answer—5050! How did he do it so fast?

He noticed a pattern. He paired numbers from the start and end:
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

Each pair adds up to the same number (101). Since there are 100 numbers, there are 50 such pairs. So, instead of adding everything one by one, he simply did:
50 × 101 = 5050

That’s the power of smart thinking!

We use the same idea in Arithmetic Progression. Instead of adding all terms individually, we pair the first and last term, second and second-last term, and so on. Each pair gives the same sum.

This “pairing trick” helps us create a formula to find the sum quickly—even for 100 or 1000 terms. So, instead of doing long calculations, we use logic and patterns to save time and effort.

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Formula for Sum of AP

Using the pairing idea, mathematicians derived a formula to find the sum of the first n terms of an AP.

Main Formula:

Sn = n/2 [2a + (n − 1)d]

Here,

• Sn = sum of first n terms
• a = first term
• d = common difference
• n = number of terms 

This formula is useful when you know a, d, and n.

Alternative Formula:

Sn = n/2 (a + l)

Here,

• l = last term 

This version is helpful when you already know the first and last terms, but not the common difference.

When to use which?

• Use Sn = n/2 [2a + (n − 1)d] when you know a, d, and n
• Use Sn = n/2 (a + l) when you know a and last term (l) 

Simple Explanation:

Think of it like this—
You’re finding the average of the first and last term, and then multiplying it by the number of terms.

This makes calculations much faster and easier compared to adding each number manually.

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Solved Examples – Sum of AP

Let’s understand how to apply the formulas with different types of questions.

Example 1: Find sum of first n terms

Find the sum of first 10 terms of AP: 2, 5, 8…

• a = 2  
• d = 3  
• n = 10 

Sn = n/2 [2a + (n − 1)d]
= 10/2 [2×2 + 9×3]
= 5 [4 + 27]
= 5 × 31
= 155

Example 2: Given sum, find n

Find number of terms if sum = 100, AP: 2, 4, 6…

• a = 2  
• d = 2  

Sn = n/2 [2a + (n − 1)d]

100 = n/2 [4 + 2(n − 1)]
100 = n/2 [2n + 2]
100 = n(n + 1)

n² + n − 100 = 0
n = 10

So, there are 10 terms.

Example 3: Real-life problem (Savings)

A child saves ₹50 in the first month and increases savings by ₹20 each month. Find total savings in 12 months.

• a = 50
• d = 20
• n = 12 

Sn = 12/2 [2×50 + 11×20]
= 6 [100 + 220]
= 6 × 320
= ₹1920

Example 4: Mixed concept problem

Find sum of AP: 5, 10, 15… up to 20th term.

• a = 5
• d = 5
• n = 20 

Sn = 20/2 [2×5 + 19×5]
= 10 [10 + 95]
= 10 × 105
= 1050

These examples show how the sum formula works in different situations. With practice, you’ll be able to solve even long problems quickly and confidently.

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Practice Questions – Sum of AP 

Level 1: Direct Formula (Find Sn)

1. Find the sum of first 10 terms of AP: 3, 7, 11…
2. Find S₁₅ for AP: 2, 5, 8…
3. Find the sum of first 20 terms of AP: 1, 4, 7…
4. Find S₁₂ for AP: 10, 8, 6…
5. Find the sum of first 25 terms of AP: 5, 10, 15…
6. Find S₈ for AP: 6, 9, 12…
7. Find the sum of first 30 terms of AP: 4, 6, 8…
8. Find S₁₈ for AP: 7, 14, 21…
9. Find the sum of first 50 terms of AP: 1, 2, 3…
10. Find S₂₀ for AP: 9, 6, 3… 

Level 2: Application (Find number of terms, missing values)

1. Find the number of terms if Sₙ = 210 for AP: 2, 4, 6…
2. Find n if sum is 300 for AP: 5, 10, 15…
3. Find the last term if S₁₀ = 155 and a = 2
4. Find d if S₁₀ = 110 and a = 5
5. Find n if Sₙ = 406 for AP: 4, 10, 16…
6. Find the first term if S₁₅ = 600 and d = 5
7. Find d if S₂₀ = 400 and a = 2
8. Find n if Sₙ = 1000 for AP: 3, 6, 9…
9. Find the last term if S₁₂ = 234 and a = 3
10. Find n if Sₙ = 528 for AP: 8, 12, 16… 

Level 3: Word Problems (Savings, salary, distance)

1. A person saves ₹100 in the first month and increases savings by ₹50 monthly. Find total savings in 12 months.
2. Salary starts at ₹5000 and increases by ₹500 yearly. Find total salary earned in 10 years.
3. A runner increases distance by 100 m daily starting from 500 m. Find total distance in 7 days.
4. A child saves ₹20 in the first week and increases by ₹10 weekly. Find total savings in 15 weeks.
5. A company produces 100 items initially and increases production by 20 each month. Find total production in 12 months.
6. A taxi fare starts at ₹10 and increases ₹5 per km. Find total fare for 10 km.
7. A ladder has 10 steps decreasing uniformly. Total length is required—frame as AP and find sum.
8. A person walks 2 km on day 1 and increases by 0.5 km daily. Find total distance in 10 days.
9. A student reads 5 pages on day 1 and increases by 2 pages daily. Find total pages read in 20 days.
10. A farmer plants rows with decreasing plants: 50, 48, 46… Find total plants in 10 rows. 

Real-Life Applications of Arithmetic Progression

Arithmetic Progression is not just a classroom concept—it appears in many real-life situations. Here are the key applications in simple pointers:

• Salary Increments:
  Many jobs offer a fixed yearly increase (e.g., ₹5000 every year), forming a clear AP pattern.
• Savings Plans:
  Saving a fixed extra amount every month (₹100, ₹200, ₹300…) follows an AP and helps in financial planning.
• Construction & Design:
  Stairs, ladders, and stadium seating often follow uniform increases or decreases in size, forming an AP.
• Business Growth:
  Production levels or sales targets that increase steadily over time can follow an AP pattern.
• Distance & Travel:
  If distance covered increases daily by a fixed amount, it forms an AP (useful in tracking progress).
• Competitive Exams:
  AP is an important topic in exams like boards and entrance tests, helping test logical thinking and problem-solving. 

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Common Mistakes Students Make in AP

Even though AP is simple, students often make avoidable mistakes. Here are the most common ones:

• Confusing the Sign of d (+/−):
  Forgetting that decreasing sequences have a negative common difference.
• Using the Wrong Formula:
  Applying the sum formula when the question asks for the nth term, or vice versa.
• Mixing Formulas:
  Getting confused between
  • an = a + (n − 1)d
  • Sn = n/2 [2a + (n − 1)d]
• Calculation Errors:
  Small arithmetic mistakes leading to incorrect answers, especially in exams.
• Not Checking if it’s an AP:
  Applying formulas without confirming whether the sequence has a constant difference.
• Incorrect Subtraction Order:
  Finding d using wrong order (always do next term − previous term). 

Avoiding these mistakes will help you solve questions faster and with better accuracy.

Quick Revision Notes 

Before exams, a quick revision can make all the difference. Here are the most important points to remember from Arithmetic Progression.

Key Formulas:

• nth term: an = a + (n − 1)d
• Sum of n terms: Sn = n/2 [2a + (n − 1)d]
• Alternate sum formula: Sn = n/2 (a + l) 

Key Concepts:

• AP = sequence with constant difference
• d = difference between consecutive terms
• Can be increasing, decreasing, or constant 

Short Tricks:

• Always find a and d first
• Use nth term formula for position-based questions
• Use sum formula when adding multiple terms 

When to Use Which Formula:

• Use an formula → when finding a specific term
• Use Sn formula → when finding total sum
• Use Sn = n/2 (a + l) → when last term is given 

Revise these points regularly, and you’ll be fully prepared to solve any AP question confidently in your exams.

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These features make learning not just easier, but more engaging, structured, and confidence-building for students.

Mastering Arithmetic Progressions Made Simple

Arithmetic Progressions become easy once you understand the basics—identifying patterns, using the nth term formula, and applying the sum formula correctly. The key is to focus on concepts rather than memorization. Remember, AP is easy once you start seeing patterns everywhere. With regular practice and the right approach, you can solve questions quickly and accurately. Stay consistent, keep practicing, and soon this chapter will feel effortless. Mastering Arithmetic Progressions means stronger confidence and better marks in your exams.

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