CBSE Class 11 Maths Revision Notes Chapter 8 Sequences and Series

Sequences and Series explains how numbers follow patterns and how their terms are added. For CBSE Class 11 Maths 2026–27, this chapter covers sequences, series, AP, GP, AM, GM and special series.

Sequences and Series is an important chapter in Class 11 Mathematics. It helps students understand ordered lists of numbers, their patterns and the sums formed from them. For example, 2, 4, 6, 8, ... is a sequence because the numbers follow a fixed order. When these terms are added as 2 + 4 + 6 + 8 + ..., they form a series.

Use these CBSE Class 11 Maths Revision Notes Chapter 8 to revise sequence, series, finite and infinite sequence, nth term, arithmetic progression, geometric progression, arithmetic mean, geometric mean and special series.

These Class 11 Maths Chapter 8 Notes are useful when you want quick revision of definitions, formulas and solved examples before practice.

Key Takeaways

• Sequence: A sequence is an ordered list of numbers.
• Series: A series is formed by adding the terms of a sequence.
• Arithmetic Progression: In an AP, the difference between consecutive terms is constant.
• Geometric Progression: In a GP, the ratio of consecutive terms is constant.

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Class 11 Maths Chapter 8 Notes for Sequences and Series Revision

These notes are arranged for 30-minute revision, so students can revise the chapter in a simple order.

Start with the meaning of a sequence. Then revise finite and infinite sequences, general term and series. After that, move to arithmetic progression, geometric progression, arithmetic mean, geometric mean and special series.

What is a Sequence?

A sequence is an ordered list of numbers.

Each number in a sequence is called a term.

Examples:

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

1, 3, 5, 7, 9, ...

5, 10, 20, 40, ...

In a sequence, order matters. The first term, second term, third term and other terms have fixed positions.

The terms of a sequence are usually written as:

a1, a2, a3, ..., an

Here, an is the nth term or general term of the sequence.

Examples of Sequences in Daily Life

Sequences appear in many real-life situations.

This is why Sequences and Series Class 11 Notes are important for later topics in Maths, science and finance.

Terms of a Sequence

The numbers in a sequence are called terms.

Example:

For the sequence:

3, 6, 9, 12, 15, ...

a1 = 3
a2 = 6
a3 = 9
a4 = 12
a5 = 15

Here, a1 is the first term and a5 is the fifth term.

nth Term or General Term

The nth term gives a formula for finding any term of a sequence.

Example:

For the sequence:

2, 4, 6, 8, ...

The nth term is:

an = 2n

Now:

a1 = 2(1) = 2
a2 = 2(2) = 4
a3 = 2(3) = 6

So, an = 2n gives the general term of the sequence.

Finite and Infinite Sequences

Sequences can be finite or infinite.

Example of finite sequence:

1, 2, 3, 4, 5

This sequence has 5 terms.

Example of infinite sequence:

1, 2, 3, 4, 5, ...

This sequence continues without ending.

Fibonacci Sequence

The Fibonacci sequence is a special sequence.

It starts as:

1, 1, 2, 3, 5, 8, 13, ...

Here, each term after the first two terms is the sum of the previous two terms.

a1 = 1
a2 = 1
an = an-1 + an-2, where n > 2

Example:

a3 = a2 + a1 = 1 + 1 = 2
a4 = a3 + a2 = 2 + 1 = 3
a5 = a4 + a3 = 3 + 2 = 5

What is a Series?

A series is formed when the terms of a sequence are added.

If a sequence is:

a1, a2, a3, ..., an

Then the corresponding series is:

a1 + a2 + a3 + ... + an

Example:

Sequence:

2, 4, 6, 8

Series:

2 + 4 + 6 + 8 = 20

A sequence lists the terms. A series adds the terms.

Sequence vs Series

This difference is one of the first ideas students should revise in Class 11 Maths Sequences and Series Notes.

Sigma Notation

Sigma notation is used to write a series in compact form.

The Greek letter Σ means “sum”.

Example:

a1 + a2 + a3 + ... + an

can be written as:

Σ ak, where k = 1 to n

Another example:

1 + 2 + 3 + ... + n

can be written as:

Σ k, where k = 1 to n

Sigma notation saves space when a series has many terms.

Arithmetic Progression

An Arithmetic Progression, or AP, is a sequence in which the difference between consecutive terms is constant.

This constant difference is called the common difference.

Example:

2, 5, 8, 11, 14, ...

Here:

5 - 2 = 3
8 - 5 = 3
11 - 8 = 3

So, the common difference is 3.

This sequence is an AP.

Common Difference in AP

If a1, a2, a3, ... is an AP, then:

d = a2 - a1 = a3 - a2 = a4 - a3

Here, d is the common difference.

Example:

For 10, 7, 4, 1, ...

d = 7 - 10 = -3

So, the common difference is -3.

An AP can have a positive, negative or zero common difference.

General Term of an AP

If the first term of an AP is a and common difference is d, then the nth term is:

an = a + (n - 1)d

Here:

a = first term
d = common difference
n = term number
an = nth term

Example:

Find the 10th term of the AP 3, 7, 11, 15, ...

Here:

a = 3
d = 4
n = 10

a10 = a + (10 - 1)d

= 3 + 9(4)

= 3 + 36

= 39

So, the 10th term is 39.

Sum of n Terms of an AP

The sum of first n terms of an AP is:

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

Another form is:

Sn = n/2 (a + l)

Here:

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

Example:

Find the sum of first 10 terms of the AP 2, 4, 6, 8, ...

Here:

a = 2
d = 2
n = 10

S10 = 10/2 [2(2) + (10 - 1)2]

= 5 [4 + 18]

= 5 × 22

= 110

So, the sum is 110.

Arithmetic Mean

The Arithmetic Mean, or AM, of two numbers a and b is:

AM = (a + b)/2

Example:

Find the AM of 6 and 10.

AM = (6 + 10)/2

= 16/2

= 8

So, the arithmetic mean is 8.

If A is the AM of a and b, then a, A, b are in AP.

Geometric Progression

A Geometric Progression, or GP, is a sequence in which the ratio of consecutive terms is constant.

This constant ratio is called the common ratio.

Example:

2, 4, 8, 16, 32, ...

Here:

4/2 = 2
8/4 = 2
16/8 = 2

So, the common ratio is 2.

This sequence is a GP.

Common Ratio in GP

If a1, a2, a3, ... is a GP, then:

r = a2/a1 = a3/a2 = a4/a3

Here, r is the common ratio.

Example:

For 81, 27, 9, 3, ...

r = 27/81 = 1/3

So, the common ratio is 1/3.

General Term of a GP

If the first term of a GP is a and common ratio is r, then the nth term is:

an = ar^(n - 1)

Here:

a = first term
r = common ratio
n = term number
an = nth term

Example:

Find the 8th term of the GP 3, 6, 12, 24, ...

Here:

a = 3
r = 2
n = 8

a8 = ar^(8 - 1)

= 3 × 2^7

= 3 × 128

= 384

So, the 8th term is 384.

Sum of n Terms of a GP

If a is the first term and r is the common ratio, then the sum of first n terms of a GP is:

Sn = a(1 - r^n)/(1 - r), if r ≠ 1

This can also be written as:

Sn = a(r^n - 1)/(r - 1), if r > 1

If r = 1, then:

Sn = na

Example:

Find the sum of first 5 terms of the GP 2, 4, 8, ...

Here:

a = 2
r = 2
n = 5

S5 = a(r^n - 1)/(r - 1)

= 2(2^5 - 1)/(2 - 1)

= 2(32 - 1)

= 62

So, the sum is 62.

Infinite Geometric Series

An infinite GP has a sum only when |r| < 1.

The sum to infinity is:

S∞ = a/(1 - r)

Here:

a = first term
r = common ratio
|r| < 1

Example:

Find the sum of:

1 + 1/2 + 1/4 + 1/8 + ...

Here:

a = 1
r = 1/2

S∞ = a/(1 - r)

= 1/(1 - 1/2)

= 1/(1/2)

= 2

So, the sum to infinity is 2.

Geometric Mean

The Geometric Mean, or GM, of two positive numbers a and b is:

GM = √ab

Example:

Find the GM of 4 and 16.

GM = √(4 × 16)

= √64

= 8

So, the geometric mean is 8.

If G is the GM of a and b, then a, G, b are in GP.

Relationship Between AM and GM

For two positive numbers a and b:

AM = (a + b)/2

GM = √ab

The relationship between AM and GM is:

AM ≥ GM

Example:

For 4 and 16:

AM = (4 + 16)/2 = 10

GM = √(4 × 16) = 8

So:

AM ≥ GM

10 ≥ 8

Harmonic Mean

The Harmonic Mean, or HM, of two positive numbers a and b is:

HM = 2ab/(a + b)

Example:

Find the HM of 4 and 6.

HM = 2ab/(a + b)

= 2(4)(6)/(4 + 6)

= 48/10

= 24/5

So, the harmonic mean is 24/5.

Arithmetic Mean, Geometric Mean and Harmonic Mean

For positive numbers:

AM ≥ GM ≥ HM

These formulas are important in Class 11 Mathematics Revision Notes Chapter 8.

Special Series

Some series are used often in Sequences and Series questions.

Sum of First n Natural Numbers

1 + 2 + 3 + ... + n = n(n + 1)/2

Example:

1 + 2 + 3 + ... + 10

= 10(10 + 1)/2

= 55

Sum of Squares of First n Natural Numbers

1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6

Example:

1² + 2² + 3² + 4² + 5²

= 5(6)(11)/6

= 55

Sum of Cubes of First n Natural Numbers

1³ + 2³ + 3³ + ... + n³ = [n(n + 1)/2]²

Example:

1³ + 2³ + 3³ + 4³

= [4(5)/2]²

= 10²

= 100

Important Formulas in Sequences and Series

Solved Examples on Sequences and Series

Example 1: Find First Three Terms

Find the first three terms of the sequence an = 2n + 5.

Solution:

For n = 1:

a1 = 2(1) + 5 = 7

For n = 2:

a2 = 2(2) + 5 = 9

For n = 3:

a3 = 2(3) + 5 = 11

So, the first three terms are:

7, 9, 11

Example 2: Find the 20th Term

Find the 20th term of the sequence an = 4n - 3.

Solution:

an = 4n - 3

a20 = 4(20) - 3

= 80 - 3

= 77

So, the 20th term is 77.

Example 3: Find the nth Term of an AP

Find the 15th term of the AP 5, 9, 13, 17, ...

Solution:

a = 5
d = 4
n = 15

an = a + (n - 1)d

a15 = 5 + (15 - 1)4

= 5 + 56

= 61

So, the 15th term is 61.

Example 4: Find the Sum of an AP

Find the sum of first 20 terms of the AP 3, 6, 9, 12, ...

Solution:

a = 3
d = 3
n = 20

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

S20 = 20/2 [2(3) + (20 - 1)3]

= 10 [6 + 57]

= 10 × 63

= 630

So, the sum is 630.

Example 5: Find the nth Term of a GP

Find the 6th term of the GP 2, 6, 18, 54, ...

Solution:

a = 2
r = 3
n = 6

an = ar^(n - 1)

a6 = 2 × 3^5

= 2 × 243

= 486

So, the 6th term is 486.

Example 6: Find the Sum of a GP

Find the sum of first 4 terms of the GP 5, 10, 20, 40, ...

Solution:

a = 5
r = 2
n = 4

S4 = a(r^n - 1)/(r - 1)

= 5(2^4 - 1)/(2 - 1)

= 5(16 - 1)

= 75

So, the sum is 75.

Common Mistakes in Sequences and Series

Quick Highlights of CBSE Class 11 Maths Notes Chapter 8

Important Terms from CBSE Class 11 Maths Revision Notes Chapter 8

The terms below cover the main definitions students need while revising this chapter.

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