Arithmetic Progressions Class 10 Full Chapter Explanation with Formulas and Questions

Arithmetic Progressions is one of the most important chapters in Class 10 Maths because it combines concept clarity, formulas, pattern recognition, and board-style application questions. It is a chapter where students can score very well if they clearly understand the meaning of a sequence, the role of common difference, the nth term formula, and the sum of n terms.

Many students find Arithmetic Progressions easy in the beginning but get confused when questions ask for a particular term, the position of a term, the last term, or the sum of several terms. That is why proper chapter understanding is important. Once the logic becomes clear, most questions from this chapter become direct and manageable.

At Deeksha Vedantu, we always encourage students to understand the pattern behind formulas instead of memorising them without meaning. Arithmetic Progressions becomes simple when students know how the sequence grows and how each formula is used.

Arithmetic Progressions at a Glance

This chapter becomes easier when students first see the main ideas together.

Quick Concept Table

What Is Arithmetic Progression

Arithmetic Progression, also called AP, is a sequence of numbers in which the difference between any two consecutive terms remains the same.

Simple Meaning of AP

If numbers are written in order and the gap between one term and the next term is constant, then that sequence is called an Arithmetic Progression.

Examples of AP

• 4, 6, 8, 10
• 5, 10, 15, 20
• -3, -6, -9, -12

In all these examples, the difference between consecutive terms is the same.

What Is Common Difference

The fixed difference between two consecutive terms of an Arithmetic Progression is called the common difference.

It is usually represented by the letter d.

Formula for Common Difference

d = succeeding term – preceding term

In standard AP notation:

d = a₂ – a₁

d = a₃ – a₂

d = a₄ – a₃

Common Difference Examples Table

Important Note

Do not find the difference between terms that are not consecutive. Common difference is always calculated between neighbouring terms.

How to Check Whether a Sequence Is an AP or Not

To check whether a sequence is an Arithmetic Progression, students should compare consecutive terms.

Step 1

Take consecutive pairs of terms.

Step 2

Find the difference between them.

Step 3

If the difference is the same throughout, then the sequence is an AP.

AP Check Table

Real-Life Examples of Arithmetic Progression

Arithmetic Progression is not just a textbook idea. It appears in daily life as well.

Real-Life AP Table

Terms of an Arithmetic Progression

In an AP, each number is called a term.

Naming the Terms

In most Class 10 questions, the first term is represented simply by a.

General Form of an Arithmetic Progression

If the first term is a and the common difference is d, then the AP can be written as:

a, a + d, a + 2d, a + 3d, …

This is the standard form of an Arithmetic Progression.

nth Term of an Arithmetic Progression

The nth term means the term at the nth position.

This is very important in board exams because students are often asked:

• find the 10th term
• find the 21st term
• which term is equal to a certain value

Formula for nth Term

aₙ = a + (n – 1)d

Where:

• a = first term
• d = common difference
• n = position of the term
• aₙ = nth term

Why the Formula Works

If you want:

• second term, then you add 1d
• third term, then you add 2d
• fourth term, then you add 3d

So for the nth term, you add (n – 1)d.

Example of Finding a Specific Term

Consider the AP:

8, 10, 12, 14, …

Find the 6th Term

Given

• a = 8
• d = 2
• n = 6

Formula

aₙ = a + (n – 1)d

Solution

a₆ = 8 + (6 – 1) × 2

a₆ = 8 + 10

a₆ = 18

Answer

The 6th term is 18.

Example: Which Term of an AP Is Zero

If in an AP, first term is 15 and common difference is -3, find which term is zero.

Given

• a = 15
• d = -3
• aₙ = 0

Formula

aₙ = a + (n – 1)d

Solution

0 = 15 + (n – 1)(-3)

0 = 15 – 3(n – 1)

3(n – 1) = 15

n – 1 = 5

n = 6

Answer

The 6th term is zero.

Example: Which Term Is 120 More Than the 21st Term

Consider the AP:

3, 15, 27, 39, …

Find which term is 120 more than its 21st term.

Step 1: Identify a and d

• a = 3
• d = 12

Step 2: Form the Equation

Let the required term be the nth term.

Then:

aₙ = a₂₁ + 120

Step 3: Use nth Term Formula

aₙ = a + (n – 1)d

a₂₁ = a + 20d

So:

a + (n – 1)d = a + 20d + 120

Subtract a from both sides:

(n – 1)d = 20d + 120

Put d = 12:

12(n – 1) = 240 + 120

12(n – 1) = 360

n – 1 = 30

n = 31

Answer

The 31st term is 120 more than the 21st term.

Example: Show That the 29th Term Is Twice the 19th Term

If the 9th term of an AP is zero, show that its 29th term is twice its 19th term.

Step 1: Use the 9th Term Information

Given:

a₉ = 0

Using the formula:

a₉ = a + 8d = 0

So:

a = -8d

Step 2: Find the 29th Term

a₂₉ = a + 28d

Put a = -8d:

a₂₉ = -8d + 28d = 20d

Step 3: Find the 19th Term

a₁₉ = a + 18d

Put a = -8d:

a₁₉ = -8d + 18d = 10d

Step 4: Compare

2 × a₁₉ = 2 × 10d = 20d

But a₂₉ = 20d

So:

a₂₉ = 2a₁₉

Answer

Hence proved.

Term from the End in an AP

Sometimes the question asks for a term from the end, not from the beginning.

Important Idea

When the term is counted from the end, the last term becomes the first term for that calculation.

Example: Term from the End

In the AP:

20, 13, 6, -1, … , -148

Find the 13th term from the end.

Step 1: Start from the End

From the end:

• first term = -148

Step 2: Find New Common Difference

Moving backward in the AP, the common difference becomes +7.

Step 3: Apply nth Term Formula

For the 13th term from the end:

a₁₃ = -148 + (13 – 1) × 7

a₁₃ = -148 + 84

a₁₃ = -64

Answer

The 13th term from the end is -64.

Sum of First n Terms of an AP

This is one of the most important parts of the chapter.

If students know the first term, common difference, and number of terms, they can find the sum easily.

Formula for Sum of First n Terms

Sₙ = n/2 [2a + (n – 1)d]

Where:

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

Alternative Sum Formula

If the last term is known, then use:

Sₙ = n/2 (a + l)

Where:

• a = first term
• l = last term
• n = number of terms

When to Use Which Sum Formula

Example: Sum of 10 Terms of an AP

Find the sum of the AP:

34, 32, 30, … up to 10 terms.

Step 1: Identify Values

• a = 34
• d = -2
• n = 10

Step 2: Apply Formula

S₁₀ = 10/2 [2 × 34 + (10 – 1)(-2)]

S₁₀ = 5 [68 – 18]

S₁₀ = 5 × 50

S₁₀ = 250

Answer

The sum of the first 10 terms is 250.

Example: Sum of 12 Terms of an AP with Negative First Term

Find the sum of the AP:

-37, -33, -29, … up to 12 terms.

Step 1: Identify Values

• a = -37
• d = 4
• n = 12

Step 2: Apply Formula

S₁₂ = 12/2 [2(-37) + (12 – 1) × 4]

S₁₂ = 6 [-74 + 44]

S₁₂ = 6 × (-30)

S₁₂ = -180

Answer

The sum of the first 12 terms is -180.

How to Find Number of Terms When Last Term Is Given

Sometimes students are given an AP with its last term and asked to find the sum. In that case, they first need the number of terms.

Example: Find the Number of Terms First

Suppose the AP is:

500, 520, 540, … , 600

Given

• a = 500
• d = 20
• l = 600

Step 1: Find Number of Terms

Use:

aₙ = a + (n – 1)d

600 = 500 + (n – 1) × 20

100 = (n – 1) × 20

n – 1 = 5

n = 6

Step 2: Use Sum Formula

Sₙ = n/2 (a + l)

S₆ = 6/2 (500 + 600)

S₆ = 3 × 1100

S₆ = 3300

Answer

The sum of the AP is 3300.

Most Important Formulas from Arithmetic Progressions

Students should memorise these formulas properly.

Formula Summary Table

Common Mistakes Students Make in Arithmetic Progressions

Common Mistakes Table

Board Exam Question Types from Arithmetic Progressions

This chapter is important because many different types of questions are asked.

Question Types Table

Study Strategy for Arithmetic Progressions

This chapter becomes easy when students revise it properly.

Step-by-Step Strategy Table

Quick Revision Notes for Students

Here is a compact revision checklist.

Revision Checklist Table

Practice Questions for Students

Important Practice Questions

• Find the common difference of 17, 13, 9, 5, …
• Find the 15th term of the AP 6, 10, 14, 18, …
• Which term of the AP 5, 9, 13, 17, … is 89?
• Find the sum of the first 20 terms of the AP 3, 7, 11, …
• Find the 10th term from the end of the AP 100, 95, 90, … , 5

FAQs

Q1. What is Arithmetic Progression in Class 10 Maths?

Arithmetic Progression is a sequence of numbers in which the difference between consecutive terms remains the same.

Q2. What is the common difference in an AP?

The common difference is the difference between a succeeding term and its preceding term.

Q3. What is the formula for the nth term of an AP?

The formula for the nth term is aₙ = a + (n – 1)d.

Q4. What is the formula for the sum of first n terms of an AP?

The formula is Sₙ = n/2 [2a + (n – 1)d].

Q5. When do we use Sₙ = n/2 (a + l)?

We use this formula when the first term, last term, and number of terms are known.

Q6. How do I find whether a sequence is an AP or not?

Find the difference between each pair of consecutive terms. If the difference stays the same throughout, then the sequence is an AP.

Q7. Why do students get confused in AP questions?

Students usually get confused because of wrong subtraction while finding d, forgetting (n – 1), or using the wrong sum formula.

Q8. Is Arithmetic Progression important for board exams?

Yes. Arithmetic Progression is a very important chapter for board exams because questions from this chapter are frequently asked in one mark, three mark, and long-answer formats.

Conclusion

Arithmetic Progressions is a foundational and scoring chapter in Class 10 Maths. Once students understand the meaning of a sequence, the idea of common difference, the nth term formula, and the sum formulas, the chapter becomes much easier than it first appears. Most mistakes in this chapter come not from difficulty, but from carelessness in signs, wrong formula use, or weak understanding of what the question is asking.

The best way to master AP is to combine concept clarity with repeated question practice. At Deeksha Vedantu, we always encourage students to revise formulas, solve different varieties of questions, and focus on understanding the logic behind each step. That approach makes Arithmetic Progressions both manageable and scoring.