CBSE Class 10 Maths Revision Notes Chapter 5 Arithmetic Progressions 2026–27

Arithmetic Progressions are number patterns where each term is obtained by adding a fixed number to the previous term.
For CBSE Class 10 Maths, Chapter 5 covers common difference, nth term, sum of terms and arithmetic mean.

Arithmetic Progressions help students study number patterns where the difference between consecutive terms remains the same. In Chapter 5, students learn how to identify an AP, find any term and calculate the sum of a fixed number of terms.

Use these CBSE Class 10 Maths Revision Notes Chapter 5 to revise the chapter step by step. Start with the meaning of AP, then move to common difference, general form, nth term, sum formulas and solved examples.

Key Takeaways

• Arithmetic Progression: A list of numbers where each term is formed by adding a fixed number to the previous term.
• Common Difference: The fixed difference between two consecutive terms.
• nth Term: an = a + (n - 1)d.
• Sum Formula: Sn = n/2 [2a + (n - 1)d].

Finding it difficult to choose the right AP formula in word problems?
Revise Chapter 5 with interactive practice, live doubt-solving sessions, and complete NCERT solutions on the Extramarks Learning App. Sign Up Free

CBSE Class 10 Maths Chapter 5 Arithmetic Progressions at a Glance

Arithmetic Progressions Class 10 focuses on sequences with a fixed difference. The chapter helps students solve questions on terms, sums and patterns from real-life situations.

Arithmetic Progression Definition for Class 10

An arithmetic progression is a list of numbers in which each term, except the first, is obtained by adding a fixed number to the previous term. This fixed number is called the common difference.

Examples of AP:

• 1, 2, 3, 4, ...
• 100, 70, 40, 10, ...
• -3, -2, -1, 0, ...
• 3, 3, 3, 3, ...

In each example, the difference between consecutive terms remains the same.

Copyable format:

Arithmetic Progression = a list of numbers with a fixed common difference.

Common Difference in Arithmetic Progression

The common difference tells whether a sequence is an AP. It is found by subtracting any term from the next term.

Copyable format:

d = a2 - a1

d = a3 - a2

d = ak+1 - ak

Here, d is the common difference.

Example:

For the AP:

6, 9, 12, 15, ...

d = 9 - 6

d = 3

Check again:

12 - 9 = 3

15 - 12 = 3

Since the difference is same, the given sequence is an AP.

Positive, Negative and Zero Common Difference

The common difference can be positive, negative or zero. This decides whether the AP increases, decreases or stays constant.

A common mistake is to subtract in the wrong order. Always subtract the previous term from the next term.

Example:

For 10, 7, 4, ...

d = 7 - 10

d = -3

The common difference is -3, not 3.

General Form of AP

The general form of AP shows how every term is built from the first term and common difference. If the first term is a and the common difference is d, the AP is written as:

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

Copyable format:

General form of AP = a, a + d, a + 2d, a + 3d, ...

Here:

• a = first term
• d = common difference
• a + d = second term
• a + 2d = third term
• a + 3d = fourth term

Example:

If a = 2 and d = 3, then the AP is:

2, 5, 8, 11, ...

Finite AP and Infinite AP

An AP can have a fixed number of terms or continue without ending. This gives two types: finite AP and infinite AP.

Finite AP

A finite AP has a limited number of terms. It also has a last term.

Example:

1, 3, 5, 7, 9

Here, the AP has 5 terms.

Infinite AP

An infinite AP has no last term. It continues without ending.

Example:

2, 4, 6, 8, 10, ...

Here, the AP continues further.

nth Term of an AP

The nth term of an AP helps students find any term without writing all previous terms. This is one of the main formulas in Chapter 5.

Copyable format:

an = a + (n - 1)d

Here:

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

Example:

Find the 10th term of the AP:

2, 7, 12, ...

Here:

a = 2

d = 7 - 2 = 5

n = 10

Use:

an = a + (n - 1)d

a10 = 2 + (10 - 1)5

a10 = 2 + 9 × 5

a10 = 2 + 45

a10 = 47

Answer:

The 10th term is 47.

nth Term from the End of a Finite AP

In a finite AP, the nth term from the end can be found using the last term. This is useful when the question asks for a term counted backwards.

Copyable format:

nth term from the end = l - (n - 1)d

Here:

• l = last term
• n = position from the end
• d = common difference of the original AP

Example:

Find the 3rd term from the end of:

4, 7, 10, 13, 16

Here:

l = 16

d = 3

n = 3

3rd term from the end = l - (n - 1)d

3rd term from the end = 16 - (3 - 1)3

3rd term from the end = 16 - 6

3rd term from the end = 10

Answer:

The 3rd term from the end is 10.

Sum of First n Terms of an AP

The sum of first n terms of an AP helps students add a fixed number of terms quickly. It avoids writing and adding every term one by one.

When first term and common difference are known:

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

When first term and last term are known:

Sn = n/2 (a + l)

Copyable format:

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

Sn = n/2 (a + l)

Here:

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

Example:

Find the sum of the first 22 terms of the AP:

8, 3, -2, ...

Here:

a = 8

d = 3 - 8 = -5

n = 22

Use:

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

S22 = 22/2 [2(8) + (22 - 1)(-5)]

S22 = 11 [16 + 21(-5)]

S22 = 11 [16 - 105]

S22 = 11(-89)

S22 = -979

Answer:

The sum of the first 22 terms is -979.

Arithmetic Mean in AP

The arithmetic mean is the middle value when three numbers are in AP. If x, y and z are in AP, then y is the arithmetic mean of x and z.

Copyable format:

y = (x + z)/2

2y = x + z

Example:

If 4, y, 10 are in AP, then:

y = (4 + 10)/2

y = 14/2

y = 7

Answer:

The arithmetic mean is 7.

Useful Properties of Arithmetic Progressions

These properties help students solve AP questions faster. Use them after revising the basic formulas.

• If the same number is added to each term of an AP, the new sequence is also an AP.
• If the same number is subtracted from each term of an AP, the new sequence is also an AP.
• If each term of an AP is multiplied by the same non-zero number, the new sequence is also an AP.
• If each term of an AP is divided by the same non-zero number, the new sequence is also an AP.
• If three numbers x, y and z are in AP, then 2y = x + z.
• If the nth term is a linear expression in n, the sequence is an AP.

AP Formulas Class 10

This table can be used with Class 10 Mathematics Revision Notes Chapter 5 for quick formula recall before solving NCERT questions.

Solved Examples from Chapter 5 Arithmetic Progressions

Solved examples help students choose the correct formula. First identify a, d and n, then decide whether the question asks for a term or a sum.

Example 1: Find First Term and Common Difference

For the AP:

3/2, 1/2, -1/2, -3/2, ...

Find a and d.

Here:

a = 3/2

d = 1/2 - 3/2

d = -2/2

d = -1

Answer:

a = 3/2

d = -1

Example 2: Find the 10th Term of an AP

Find the 10th term of the AP:

2, 7, 12, ...

Here:

a = 2

d = 7 - 2 = 5

n = 10

Use:

an = a + (n - 1)d

a10 = 2 + (10 - 1)5

a10 = 2 + 45

a10 = 47

Answer:

The 10th term is 47.

Example 3: Check Whether a Sequence is an AP

Check whether the sequence is an AP:

4, 10, 16, 22, ...

Find consecutive differences:

10 - 4 = 6

16 - 10 = 6

22 - 16 = 6

Since the difference is same, it is an AP.

Answer:

The sequence is an AP with common difference 6.

Example 4: Find the Sum of First 20 Multiples of 6

The first 20 multiples of 6 are:

6, 12, 18, 24, ...

Here:

a = 6

d = 6

n = 20

Use:

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

S20 = 20/2 [2(6) + (20 - 1)6]

S20 = 10 [12 + 19(6)]

S20 = 10 [12 + 114]

S20 = 10 × 126

S20 = 1260

Answer:

The sum of the first 20 multiples of 6 is 1260.

Chapter 5 Arithmetic Progressions Summary

These Class 10 Maths Chapter 5 Notes help students revise AP meaning, common difference, nth term, sum formulas and arithmetic mean in one flow.

• An arithmetic progression is a list of numbers with a fixed common difference.
• The common difference is found by subtracting the previous term from the next term.
• The general form of AP is:
  • a, a + d, a + 2d, a + 3d, ...
• The nth term of an AP is:
  • an = a + (n - 1)d
• The sum of first n terms of an AP is:
  • Sn = n/2 [2a + (n - 1)d]
• If the last term is given, use:
  • Sn = n/2 (a + l)
• If x, y and z are in AP, then:
  • 2y = x + z
• A finite AP has a last term.
• An infinite AP has no last term.

Useful Important Questions Class 10 Maths Links