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Class 11 Chapter 8

Class 11 Maths Chapter 8 Sequences and Series Revision Notes

Many natural and real-world phenomena follow a predictable pattern—monthly savings, population growth, compound interest, and even the arrangement of objects often form mathematical sequences. Sequences and Series provide the tools to recognize these patterns, determine future terms, and calculate their sums efficiently. This chapter introduces Arithmetic Progressions (AP), Geometric Progressions (GP), special series, and their applications in solving practical as well as competitive examination problems. The concepts learned here are fundamental to Calculus, Probability, Finance, Physics, and Engineering, making this chapter one of the most important topics for JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have prepared these Class 11 Maths Chapter 8: Sequences and Series Revision Notes to help you master progression-based problems quickly and accurately. These notes include concise theory, important formulas, shortcut methods, solved examples, standard identities, and previous years' question analysis. Whether you're preparing for board examinations or aiming for top ranks in JEE, these revision notes will strengthen your conceptual understanding and improve your problem-solving speed.

1.0Chapter Snapshot

Topic	Overview
Sequences	Ordered collection of numbers
Series	Sum of terms of a sequence
Arithmetic Progression (AP)	Constant common difference
Arithmetic Mean (AM)	Mean between two numbers
Sum of AP	Formula for the sum of first (n) terms
Geometric Progression (GP)	Constant common ratio
Geometric Mean (GM)	Mean between two positive numbers
Sum of GP	Finite and infinite geometric series

2.0Chapter 8 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Test Paper with Solution

Test your Knowledge

question 1 of 5

The common difference of the AP 3, 7, 11, 15, ... is:

1.2

2.3

3.4

4.5

Sequence and Series

A sequence refers to a structured arrangement of elements in a specific order that follows a specific pattern or rule. These elements can be numbers, letters, or any other objects. In mathematics, sequences are fundamental concepts that are extensively used in various areas, such as calculus, algebra, and number theory.

A series represents the cumulative total obtained by adding together the terms of a sequence. Series are essential in mathematics and have wide applications in calculus, physics, engineering, and many other fields.

Understanding sequences and series is crucial in mathematics as they form the basis for calculus, which deals with concepts like limits, derivatives, and integrals. They also play a vital role in solving mathematical problems, modeling real-world phenomena, and developing mathematical theories and techniques.

1.0Sequence and Series Definition

A sequence refers to a structured arrangement of elements in a specific order that follows a specific pattern or rule. Each term in a sequence is obtained by applying a consistent rule to the preceding term. For example, the Sequence of whole numbers (0, 1, 2, 3, 4, ...) is formed by adding 1 to each preceding term.

A series represents the cumulative total obtained by adding together the terms of a sequence. It is formed by adding all the terms of a sequence together. For instance, the sum of the first n natural numbers is an example of a series, denoted as 1 + 2 + 3 + ... + n.

In summary, a sequence is a list of terms following a pattern, while a series is the sum of these terms. Both sequences and series are fundamental concepts in mathematics and are widely used in various applications, including calculus, number theory, and finance.

2.0Types of Sequence and Series

There are several types of sequences and series in mathematics, each with its own characteristics and properties. Here are some common types of Sequence

Arithmetic Sequence:

In an arithmetic sequence, each term is generated by adding a constant value (known as the common difference) to the preceding term.

Example: 2, 5, 8, 11, 14, ...

Geometric Sequence:

G. P is a sequence of non-zero numbers. Each of the succeeding terms is equal to the preceding term multiplied by a constant. Thus, in a GP, the ratio of the successive terms is constant (called the common ratio).

Example: 2, 6, 18, 54, …

Harmonic Sequence:

In a harmonic sequence, the reciprocals of the terms form an arithmetic sequence.

Example: 1, $\frac{1}{2}, \frac{1}{4}, \frac{3}{4}, ...$

Fibonacci Sequence:

The Fibonacci sequence is a distinctive series where each number is the sum of the two preceding numbers. It commences with 0 and 1, and the following terms are formed by adding the two previous numbers.

Example: 0, 1, 1, 2, 3, 5, 8, 13, and so forth.

Common Types of Series

Arithmetic Series:

An arithmetic series refers to the summation of terms within an arithmetic sequence.

Example: Sum of the first 10 terms of the sequence 2, 5, 8, 11, ...

Geometric Series:

A geometric series represents the total of terms within a geometric sequence.

Example: Sum of the first 5 terms of the sequence 2, 6, 18, 54, ...

Infinite Series:

An infinite series is a series that extends endlessly without a definite endpoint, either with a fixed pattern or following a specific rule.

Example: Sum of the infinite series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$

Convergent and Divergent Series:

A convergent series is characterized by its sum gradually approaching a finite value as more terms are added.

A divergent series is one whose sum does not approach a finite value as the number of terms increases.

1.0Difference between Sequence and Series

Sequence

A sequence refers to a structured arrangement of elements in a specific order or numbers arranged according to a rule or pattern.
It is represented as a₁, a₂, a₃, …, a_n, where a₁, a₂, a₃, … are the individual terms or elements of the sequence.
A sequence can be finite (having a specific last term) or infinite (continuing indefinitely).

Examples of sequences include arithmetic sequences, geometric sequences, Fibonacci sequences, etc.

The focus in a sequence is on the arrangement and order of the elements based on a specific rule or pattern.

Series

A series represents the cumulative total obtained by adding together the terms of a sequence.
It can be represented as S_n = a₁ + a₂ + a₃ + … + a_n, where S_n represents the sum of the initial n terms of the Sequence.
A series can also be finite (summing a specific number of terms) or infinite (summing an infinite number of terms).

Examples of series include arithmetic series, geometric series, infinite series, etc.

The focus in a series is on finding the total sum or value obtained by adding the terms of a sequence.

2.0Sequence and Series Formula

Here are some common formulas related to sequences and series:

Arithmetic Sequence:

a_n = a₁ + (n−1)⋅d - n^th term of an arithmetic sequence

$S_{n} = \frac{n}{2} \cdot (a_{1} + a_{n})$ - sum of the first n terms of an arithmetic sequence

$S_{n} = \frac{n}{2} \cdot (2 a_{1} + (n - 1) \cdot d)$ - sum of an arithmetic series

Geometric Sequence:

a_n = a₁⋅rⁿ⁻¹- n^th term of a geometric sequence

$S_{n} = a_{1} \cdot \frac{( r _{n} - 1 )}{r - 1}$ - sum of the first n terms of a geometric sequence

$S = \frac{a _{1}}{1 - r}$ - sum of an infinite geometric series (when ∣r∣<1)

Harmonic Series:

$S_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ - sum of the first n terms of the harmonic series (n is finite)

Fibonacci Sequence:

F_n = F_n−1 + F_n−2 - n^th term of the Fibonacci sequence (starting with F₁ = 1, F₂ = 1 or F₀ = 0, F₁ = 1)

3.0Sequence and Series Jee Mains Question

Let S_n denote the sum of the initial n terms of an arithmetic progression. If S₁₀= 390 and the ratio of the tenth and the fifth terms is 15:7, then S₁₅ − S₅ is equal to?

a. 690 b. 890 c. 790 d. 800

Solution: (c)

We know that in an A. P Formula for the sum of n terms is

$S_{n} = \frac{n}{2} \cdot [2 a + (n - 1) \cdot d]$

$S_{10} = \frac{10}{2} \cdot [2 a + (10 - 1) \cdot d]$

390 = 5·(2a + 9d)

390 = 10a + 45d

78 = 2a + 9d . . . (1)

Now we have given the ratio of the tenth and fifth terms as 15: 7,

Then, in an A.P., the nth term is given by a_n = a + (n−1).d

So, the Tenth term a₁₀ = a + (10−1).d

a₁₀ = a + 9d

And the Fifth term is a₅ = a + (5−1).d

a₅ = a + 4d

We know the ratio as follows:

a+ $\frac{9 d}{a}$ +4 d= $\frac{15}{7}$

7(a + 9d) = 15(a + 4d)

7a + 63d = 15a + 60d

63d – 60d = 15a –7a

3d = 8a . . . (2)

Solving Eq (1) and Eq (2)

We get a =3, and d = 8

Now, we can find S₁₅ and S₅ by using the formula of the Sum of an AP

So, S₁₅ = 885 and S₅ = 95

So, S₁₅−S₅ = 885 – 95 = 790

Therefore, the correct answer is 790, Option (c).

Let the 2^nd, 8^th, and 44^th terms of a non-constant A. P. be, respectively, the 1^st, 2^nd, and 3^rd terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to

a. 960 b. 980 c. 990 d. 970

Solution: (d)

From the given question the 2^nd, 8^th and 44^th terms of an A.P is 1 + d, 1 + 7d, and 1 + 43d.

If these terms are in G. P, then,

(1 + 7d)² = (1 + d) (1 + 43d )

1 + 49d² + 14d = 1 + 44d + 43d²

6d² – 30d = 0

d = 5

Now, using the Formula of the Sum of the first n initial terms of an A.P

$S_{n} = \frac{n}{2} \cdot [2 a + (n - 1) \cdot d]$

$S_{20} = \frac{20}{2} \cdot [2.1 + (20 - 1) \cdot 5]$

S₂₀ = 10[2 + 95]

S₂₀ =970

So, the correct answer is (d), which is 970.

4.0Sequence and Series Jee Advanced Question

Question 1: If the sum of the initial n terms of an Arithmetic Progression is cn², then the sum of the squares of these n terms is

$\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{6}$
$\frac{n ( 4 n ^{2} + 1 ) c ^{2}}{3}$
$\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{3}$
$\frac{n ( 4 n ^{2} + 1 ) c ^{2}}{6}$

Solution: (c)

Let the sum of the first n terms of an A.P be S_n.

Given S_n = cn²

S_n–1 = c (n – 1)²

= cn² – cn² + 2cn –c

= c (2n –1)

Sum of squares = ΣTn²

= c²[4Σn² + Σ1 – 4 Σn]

= $[\frac{4 n ^{2} n ( n + 1 ) ( 2 n + 1 )}{6}] + n c^{2} - [2 c^{2} n (n + 1)]$

= $\frac{n c ^{2} ( 4 n ^{2} + 6 n + 2 + 3 - 6 n - 6 )}{3}$

= $\frac{n c ^{2} ( 4 n ^{2} - 1 )}{3}$

= $\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{3}$

Hence, option(c) is the answer.

5.0Sequence and Series Solved Problem

Example 1: If a, b, c, d, and p are distinct real numbers such that the inequality (a² + b² + c²) p² – 2p(ab + bc + cd) + (b² + c² + d²) ≤ 0, then a, b, c, d are in

a. A.P b. G.P c. H.P d. None of these

Solution: (b)

Here, the given condition (a² + b² + c²)p² – 2p(ab + bc + cd) + (b² + c² +d²) ≤ 0

This implies (ap – b)² + (bp – c)² + (cp – d)² ≤ 0

Therefore, a square cannot be negative

Therefore, ap –b = 0, bp – c = 0 implies $p = \frac{b}{a} = \frac{c}{b} = \frac{d}{c}$ implies a, b, c, and dare in G.P.

Example 2: Let a, b, and c be positive integers s.t $\frac{b}{a}$ is an integer. If a, b, and c are in G. P and the arithmetic mean of a, b, and c is b + 2, then the value of $\frac{( a ^{2} + a - 14 )}{( a + 1 )}$ is

Solution:

Given that a, b, and c are in geometric progression.

Let a, b, c be a, ar, ar² (where r is the common ratio)

Given $\frac{( a + b + c )}{3}$ =b+2

Implies (a + ar + ar²) = 3(ar) + 6(since b = ar)

Implies ar² – 2ar + a = 6

⇒ a(r² – 2r + 1) = 6

⇒ a(r – 1)² = 6

⇒ (r – 1)² = $\frac{6}{a}$

$\frac{6}{a}$ should be a perfect square.

So, the possible value for a is 6.

$\frac{( a ^{2} + a - 14 )}{( a + 1 )} = \frac{( 36 + 6 - 14 )}{7} = \frac{28}{7} = 4$

Hence the value of $\frac{( a ^{2} + a - 14 )}{( a + 1 )}$ is 4

6.0Sample Question on Sequence and Series

Q. What is the sum of an infinite geometric series?

Ans: The sum of infinite geometric series can be found using the formula $S = \frac{a _{1}}{1 - r}$ , where the first term is a₁, and the common ratio between consecutive terms is r.

Also Read:-

Arithmetic and Geometric Sequences	Series	Derivative Function Calculus
Differential Calculus	Integral Calculus	Arithmetic and Geometric Sequences
Geometric Mean	Geometric Distribution	Sequences and Series PYQs

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the concepts of sequences and series.
Solve problems involving Arithmetic Progressions (AP).
Find arithmetic means between given numbers.
Calculate the sum of finite arithmetic series.
Solve Geometric Progression (GP) problems.
Find geometric means between given numbers.
Evaluate finite and infinite geometric series.
Apply progression concepts to solve JEE Main, JEE Advanced, and CBSE-level questions.

2.05-Minute Quick Revision

Must-Revise Concepts

Sequence
Series
Arithmetic Progression (AP)
Common Difference
Arithmetic Mean
Sum of AP
Geometric Progression (GP)
Common Ratio
Geometric Mean
Infinite GP

Important Formula Recall

(a_n=a+(n-1)d)
(S_n=\dfrac{n}{2}[2a+(n-1)d])
(S_n=\dfrac{n}{2}(a+l))
(a_n=ar^{,n-1})
(S_n=\dfrac{a(r^n-1)}{r-1},;r\neq1)
(S_n=\dfrac{a(1-r^n)}{1-r},;|r|<1)
(S_\infty=\dfrac{a}{1-r},;|r|<1)

Important Properties

In an AP, the difference between consecutive terms remains constant.
In a GP, the ratio of consecutive terms remains constant.
Arithmetic Mean is inserted in an AP.
Geometric Mean is inserted in a GP.
An infinite GP converges only when (|r|<1).

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Arithmetic Progression	Very High	Easy–Moderate
Sum of AP	High	Moderate
Arithmetic Mean	Medium	Easy
Geometric Progression	Very High	Moderate
Sum of GP	Very High	Moderate
Infinite GP	High	Moderate
Mixed Progression Problems	Very High	Moderate–High

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Sequences and Series in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Sequences and Series is a formula-based chapter where conceptual clarity is essential. Students often make mistakes by using incorrect formulas, confusing AP with GP, or applying infinite GP formulas under invalid conditions. Careful identification of the progression type is the key to solving questions correctly.

Common Mistake	Correct Approach
Confusing AP with GP	First check whether the terms have a common difference or a common ratio.
Using the wrong sum formula	Select the appropriate AP or GP formula based on the progression.
Applying the infinite GP formula incorrectly	Use (S_\infty=\dfrac{a}{1-r}) only when (
Incorrect calculation of common difference or ratio	Verify (d) or (r) before solving further.
Errors in finding the (n^{th}) term	Substitute values carefully into the standard formula.
Ignoring negative common ratios	Handle signs carefully while calculating GP terms.
Mixing Arithmetic Mean and Geometric Mean	Remember that AM applies to AP, whereas GM applies to GP.
Calculation mistakes in exponentiation	Simplify powers before substituting into GP formulas.

6.0ALLEN Faculty Tips

Identify whether the given sequence is an AP or a GP before applying formulas.
Memorize all standard progression formulas for quick recall.
Practice finding missing terms using both AP and GP concepts.
Revise infinite GP separately, as it is frequently tested.
Use the relation between AM and GM in conceptual questions.
Simplify expressions before substituting numerical values.
Solve Previous Years' Questions regularly to improve speed and accuracy.

7.0PYQ Trend Analysis

Sequences and Series is one of the highest-weightage Algebra chapters in JEE Main and JEE Advanced. Questions commonly involve arithmetic progressions, geometric progressions, series summation, arithmetic mean, geometric mean, and mixed application-based problems.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Arithmetic Progression	2–3	Easy–Moderate
Sum of AP	2–3	Moderate
Arithmetic Mean	1–2	Easy
Geometric Progression	2–4	Moderate
Sum of GP	2–3	Moderate
Infinite GP	1–2	Moderate
Mixed Progression Problems	2–4	Moderate–High

8.0Most Common Question Types

Finding the (n^{th}) term of an Arithmetic Progression.
Calculating the sum of the first (n) terms of an AP.
Finding Arithmetic Mean(s) between given numbers.
Solving Geometric Progression problems.
Finding the sum of finite and infinite GPs.
Determining Geometric Mean(s) between two numbers.
Solving mixed AP-GP application-based questions.
JEE Advanced problems involving progression-based reasoning.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise the concepts of sequences and series.
Memorize all AP formulas and practice finding terms.
Revise the sum formulas for Arithmetic Progressions.
Learn GP formulas and common ratio concepts.
Practice finite and infinite GP problems.
Revise Arithmetic Mean and Geometric Mean formulas.
Solve Previous Years' Questions (PYQs).
Finish with mixed practice sets involving AP, GP, and application-based questions.

Class 11 Maths Chapters List:- JEE Revision Notes

Chapter 1 Sets

Chapter 2 Relations and Functions

Chapter 3 Trigonometric Functions

Chapter 4 Complex Numbers and Quadratic Equations

Chapter 5 Linear Inequalities

Chapter 6 Permutations and Combinations

Chapter 7 Binomial Theorem

Chapter 8 Sequence and Series

Chapter 9 Straight Lines

Chapter 10 Conic Sections

Chapter 11 Introduction to Three-dimensional Geometry

Chapter 12 Limits and Derivatives

Chapter 13 Statistics

Chapter 14 Probability

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Class 11 Maths Chapter 8 Sequences and Series Revision Notes

Many natural and real-world phenomena follow a predictable pattern—monthly savings, population growth, compound interest, and even the arrangement of objects often form mathematical sequences. Sequences and Series provide the tools to recognize these patterns, determine future terms, and calculate their sums efficiently. This chapter introduces Arithmetic Progressions (AP), Geometric Progressions (GP), special series, and their applications in solving practical as well as competitive examination problems. The concepts learned here are fundamental to Calculus, Probability, Finance, Physics, and Engineering, making this chapter one of the most important topics for JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have prepared these Class 11 Maths Chapter 8: Sequences and Series Revision Notes to help you master progression-based problems quickly and accurately. These notes include concise theory, important formulas, shortcut methods, solved examples, standard identities, and previous years' question analysis. Whether you're preparing for board examinations or aiming for top ranks in JEE, these revision notes will strengthen your conceptual understanding and improve your problem-solving speed.

1.0Chapter Snapshot

Topic	Overview
Sequences	Ordered collection of numbers
Series	Sum of terms of a sequence
Arithmetic Progression (AP)	Constant common difference
Arithmetic Mean (AM)	Mean between two numbers
Sum of AP	Formula for the sum of first (n) terms
Geometric Progression (GP)	Constant common ratio
Geometric Mean (GM)	Mean between two positive numbers
Sum of GP	Finite and infinite geometric series

2.0Chapter 8 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Test Paper with Solution

Sequence and Series

A sequence refers to a structured arrangement of elements in a specific order that follows a specific pattern or rule. These elements can be numbers, letters, or any other objects. In mathematics, sequences are fundamental concepts that are extensively used in various areas, such as calculus, algebra, and number theory.

A series represents the cumulative total obtained by adding together the terms of a sequence. Series are essential in mathematics and have wide applications in calculus, physics, engineering, and many other fields.

Understanding sequences and series is crucial in mathematics as they form the basis for calculus, which deals with concepts like limits, derivatives, and integrals. They also play a vital role in solving mathematical problems, modeling real-world phenomena, and developing mathematical theories and techniques.

1.0Sequence and Series Definition

A sequence refers to a structured arrangement of elements in a specific order that follows a specific pattern or rule. Each term in a sequence is obtained by applying a consistent rule to the preceding term. For example, the Sequence of whole numbers (0, 1, 2, 3, 4, ...) is formed by adding 1 to each preceding term.

A series represents the cumulative total obtained by adding together the terms of a sequence. It is formed by adding all the terms of a sequence together. For instance, the sum of the first n natural numbers is an example of a series, denoted as 1 + 2 + 3 + ... + n.

In summary, a sequence is a list of terms following a pattern, while a series is the sum of these terms. Both sequences and series are fundamental concepts in mathematics and are widely used in various applications, including calculus, number theory, and finance.

2.0Types of Sequence and Series

There are several types of sequences and series in mathematics, each with its own characteristics and properties. Here are some common types of Sequence

Arithmetic Sequence:

In an arithmetic sequence, each term is generated by adding a constant value (known as the common difference) to the preceding term.

Example: 2, 5, 8, 11, 14, ...

Geometric Sequence:

G. P is a sequence of non-zero numbers. Each of the succeeding terms is equal to the preceding term multiplied by a constant. Thus, in a GP, the ratio of the successive terms is constant (called the common ratio).

Example: 2, 6, 18, 54, …

Harmonic Sequence:

In a harmonic sequence, the reciprocals of the terms form an arithmetic sequence.

Example: 1, $\frac{1}{2}, \frac{1}{4}, \frac{3}{4}, ...$

Fibonacci Sequence:

The Fibonacci sequence is a distinctive series where each number is the sum of the two preceding numbers. It commences with 0 and 1, and the following terms are formed by adding the two previous numbers.

Example: 0, 1, 1, 2, 3, 5, 8, 13, and so forth.

Common Types of Series

Arithmetic Series:

An arithmetic series refers to the summation of terms within an arithmetic sequence.

Example: Sum of the first 10 terms of the sequence 2, 5, 8, 11, ...

Geometric Series:

A geometric series represents the total of terms within a geometric sequence.

Example: Sum of the first 5 terms of the sequence 2, 6, 18, 54, ...

Infinite Series:

An infinite series is a series that extends endlessly without a definite endpoint, either with a fixed pattern or following a specific rule.

Example: Sum of the infinite series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$

Convergent and Divergent Series:

A convergent series is characterized by its sum gradually approaching a finite value as more terms are added.

A divergent series is one whose sum does not approach a finite value as the number of terms increases.

1.0Difference between Sequence and Series

Sequence

A sequence refers to a structured arrangement of elements in a specific order or numbers arranged according to a rule or pattern.
It is represented as a₁, a₂, a₃, …, a_n, where a₁, a₂, a₃, … are the individual terms or elements of the sequence.
A sequence can be finite (having a specific last term) or infinite (continuing indefinitely).

Examples of sequences include arithmetic sequences, geometric sequences, Fibonacci sequences, etc.

The focus in a sequence is on the arrangement and order of the elements based on a specific rule or pattern.

Series

A series represents the cumulative total obtained by adding together the terms of a sequence.
It can be represented as S_n = a₁ + a₂ + a₃ + … + a_n, where S_n represents the sum of the initial n terms of the Sequence.
A series can also be finite (summing a specific number of terms) or infinite (summing an infinite number of terms).

Examples of series include arithmetic series, geometric series, infinite series, etc.

The focus in a series is on finding the total sum or value obtained by adding the terms of a sequence.

2.0Sequence and Series Formula

Here are some common formulas related to sequences and series:

Arithmetic Sequence:

a_n = a₁ + (n−1)⋅d - n^th term of an arithmetic sequence

$S_{n} = \frac{n}{2} \cdot (a_{1} + a_{n})$ - sum of the first n terms of an arithmetic sequence

$S_{n} = \frac{n}{2} \cdot (2 a_{1} + (n - 1) \cdot d)$ - sum of an arithmetic series

Geometric Sequence:

a_n = a₁⋅rⁿ⁻¹- n^th term of a geometric sequence

$S_{n} = a_{1} \cdot \frac{( r _{n} - 1 )}{r - 1}$ - sum of the first n terms of a geometric sequence

$S = \frac{a _{1}}{1 - r}$ - sum of an infinite geometric series (when ∣r∣<1)

Harmonic Series:

$S_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ - sum of the first n terms of the harmonic series (n is finite)

Fibonacci Sequence:

F_n = F_n−1 + F_n−2 - n^th term of the Fibonacci sequence (starting with F₁ = 1, F₂ = 1 or F₀ = 0, F₁ = 1)

3.0Sequence and Series Jee Mains Question

Let S_n denote the sum of the initial n terms of an arithmetic progression. If S₁₀= 390 and the ratio of the tenth and the fifth terms is 15:7, then S₁₅ − S₅ is equal to?

a. 690 b. 890 c. 790 d. 800

Solution: (c)

We know that in an A. P Formula for the sum of n terms is

$S_{n} = \frac{n}{2} \cdot [2 a + (n - 1) \cdot d]$

$S_{10} = \frac{10}{2} \cdot [2 a + (10 - 1) \cdot d]$

390 = 5·(2a + 9d)

390 = 10a + 45d

78 = 2a + 9d . . . (1)

Now we have given the ratio of the tenth and fifth terms as 15: 7,

Then, in an A.P., the nth term is given by a_n = a + (n−1).d

So, the Tenth term a₁₀ = a + (10−1).d

a₁₀ = a + 9d

And the Fifth term is a₅ = a + (5−1).d

a₅ = a + 4d

We know the ratio as follows:

a+ $\frac{9 d}{a}$ +4 d= $\frac{15}{7}$

7(a + 9d) = 15(a + 4d)

7a + 63d = 15a + 60d

63d – 60d = 15a –7a

3d = 8a . . . (2)

Solving Eq (1) and Eq (2)

We get a =3, and d = 8

Now, we can find S₁₅ and S₅ by using the formula of the Sum of an AP

So, S₁₅ = 885 and S₅ = 95

So, S₁₅−S₅ = 885 – 95 = 790

Therefore, the correct answer is 790, Option (c).

Let the 2^nd, 8^th, and 44^th terms of a non-constant A. P. be, respectively, the 1^st, 2^nd, and 3^rd terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to

a. 960 b. 980 c. 990 d. 970

Solution: (d)

From the given question the 2^nd, 8^th and 44^th terms of an A.P is 1 + d, 1 + 7d, and 1 + 43d.

If these terms are in G. P, then,

(1 + 7d)² = (1 + d) (1 + 43d )

1 + 49d² + 14d = 1 + 44d + 43d²

6d² – 30d = 0

d = 5

Now, using the Formula of the Sum of the first n initial terms of an A.P

$S_{n} = \frac{n}{2} \cdot [2 a + (n - 1) \cdot d]$

$S_{20} = \frac{20}{2} \cdot [2.1 + (20 - 1) \cdot 5]$

S₂₀ = 10[2 + 95]

S₂₀ =970

So, the correct answer is (d), which is 970.

4.0Sequence and Series Jee Advanced Question

Question 1: If the sum of the initial n terms of an Arithmetic Progression is cn², then the sum of the squares of these n terms is

$\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{6}$
$\frac{n ( 4 n ^{2} + 1 ) c ^{2}}{3}$
$\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{3}$
$\frac{n ( 4 n ^{2} + 1 ) c ^{2}}{6}$

Solution: (c)

Let the sum of the first n terms of an A.P be S_n.

Given S_n = cn²

S_n–1 = c (n – 1)²

= cn² – cn² + 2cn –c

= c (2n –1)

Sum of squares = ΣTn²

= c²[4Σn² + Σ1 – 4 Σn]

= $[\frac{4 n ^{2} n ( n + 1 ) ( 2 n + 1 )}{6}] + n c^{2} - [2 c^{2} n (n + 1)]$

= $\frac{n c ^{2} ( 4 n ^{2} + 6 n + 2 + 3 - 6 n - 6 )}{3}$

= $\frac{n c ^{2} ( 4 n ^{2} - 1 )}{3}$

= $\frac{n ( 4 n ^{2} - 1 ) c ^{2}}{3}$

Hence, option(c) is the answer.

5.0Sequence and Series Solved Problem

Example 1: If a, b, c, d, and p are distinct real numbers such that the inequality (a² + b² + c²) p² – 2p(ab + bc + cd) + (b² + c² + d²) ≤ 0, then a, b, c, d are in

a. A.P b. G.P c. H.P d. None of these

Solution: (b)

Here, the given condition (a² + b² + c²)p² – 2p(ab + bc + cd) + (b² + c² +d²) ≤ 0

This implies (ap – b)² + (bp – c)² + (cp – d)² ≤ 0

Therefore, a square cannot be negative

Therefore, ap –b = 0, bp – c = 0 implies $p = \frac{b}{a} = \frac{c}{b} = \frac{d}{c}$ implies a, b, c, and dare in G.P.

Example 2: Let a, b, and c be positive integers s.t $\frac{b}{a}$ is an integer. If a, b, and c are in G. P and the arithmetic mean of a, b, and c is b + 2, then the value of $\frac{( a ^{2} + a - 14 )}{( a + 1 )}$ is

Solution:

Given that a, b, and c are in geometric progression.

Let a, b, c be a, ar, ar² (where r is the common ratio)

Given $\frac{( a + b + c )}{3}$ =b+2

Implies (a + ar + ar²) = 3(ar) + 6(since b = ar)

Implies ar² – 2ar + a = 6

⇒ a(r² – 2r + 1) = 6

⇒ a(r – 1)² = 6

⇒ (r – 1)² = $\frac{6}{a}$

$\frac{6}{a}$ should be a perfect square.

So, the possible value for a is 6.

$\frac{( a ^{2} + a - 14 )}{( a + 1 )} = \frac{( 36 + 6 - 14 )}{7} = \frac{28}{7} = 4$

Hence the value of $\frac{( a ^{2} + a - 14 )}{( a + 1 )}$ is 4

6.0Sample Question on Sequence and Series

Q. What is the sum of an infinite geometric series?

Ans: The sum of infinite geometric series can be found using the formula $S = \frac{a _{1}}{1 - r}$ , where the first term is a₁, and the common ratio between consecutive terms is r.

Also Read:-

Arithmetic and Geometric Sequences	Series	Derivative Function Calculus
Differential Calculus	Integral Calculus	Arithmetic and Geometric Sequences
Geometric Mean	Geometric Distribution	Sequences and Series PYQs

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the concepts of sequences and series.
Solve problems involving Arithmetic Progressions (AP).
Find arithmetic means between given numbers.
Calculate the sum of finite arithmetic series.
Solve Geometric Progression (GP) problems.
Find geometric means between given numbers.
Evaluate finite and infinite geometric series.
Apply progression concepts to solve JEE Main, JEE Advanced, and CBSE-level questions.

2.05-Minute Quick Revision

Must-Revise Concepts

Sequence
Series
Arithmetic Progression (AP)
Common Difference
Arithmetic Mean
Sum of AP
Geometric Progression (GP)
Common Ratio
Geometric Mean
Infinite GP

Important Formula Recall

(a_n=a+(n-1)d)
(S_n=\dfrac{n}{2}[2a+(n-1)d])
(S_n=\dfrac{n}{2}(a+l))
(a_n=ar^{,n-1})
(S_n=\dfrac{a(r^n-1)}{r-1},;r\neq1)
(S_n=\dfrac{a(1-r^n)}{1-r},;|r|<1)
(S_\infty=\dfrac{a}{1-r},;|r|<1)

Important Properties

In an AP, the difference between consecutive terms remains constant.
In a GP, the ratio of consecutive terms remains constant.
Arithmetic Mean is inserted in an AP.
Geometric Mean is inserted in a GP.
An infinite GP converges only when (|r|<1).

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Arithmetic Progression	Very High	Easy–Moderate
Sum of AP	High	Moderate
Arithmetic Mean	Medium	Easy
Geometric Progression	Very High	Moderate
Sum of GP	Very High	Moderate
Infinite GP	High	Moderate
Mixed Progression Problems	Very High	Moderate–High

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Sequences and Series in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Sequences and Series is a formula-based chapter where conceptual clarity is essential. Students often make mistakes by using incorrect formulas, confusing AP with GP, or applying infinite GP formulas under invalid conditions. Careful identification of the progression type is the key to solving questions correctly.

Common Mistake	Correct Approach
Confusing AP with GP	First check whether the terms have a common difference or a common ratio.
Using the wrong sum formula	Select the appropriate AP or GP formula based on the progression.
Applying the infinite GP formula incorrectly	Use (S_\infty=\dfrac{a}{1-r}) only when (
Incorrect calculation of common difference or ratio	Verify (d) or (r) before solving further.
Errors in finding the (n^{th}) term	Substitute values carefully into the standard formula.
Ignoring negative common ratios	Handle signs carefully while calculating GP terms.
Mixing Arithmetic Mean and Geometric Mean	Remember that AM applies to AP, whereas GM applies to GP.
Calculation mistakes in exponentiation	Simplify powers before substituting into GP formulas.

6.0ALLEN Faculty Tips

Identify whether the given sequence is an AP or a GP before applying formulas.
Memorize all standard progression formulas for quick recall.
Practice finding missing terms using both AP and GP concepts.
Revise infinite GP separately, as it is frequently tested.
Use the relation between AM and GM in conceptual questions.
Simplify expressions before substituting numerical values.
Solve Previous Years' Questions regularly to improve speed and accuracy.

7.0PYQ Trend Analysis

Sequences and Series is one of the highest-weightage Algebra chapters in JEE Main and JEE Advanced. Questions commonly involve arithmetic progressions, geometric progressions, series summation, arithmetic mean, geometric mean, and mixed application-based problems.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Arithmetic Progression	2–3	Easy–Moderate
Sum of AP	2–3	Moderate
Arithmetic Mean	1–2	Easy
Geometric Progression	2–4	Moderate
Sum of GP	2–3	Moderate
Infinite GP	1–2	Moderate
Mixed Progression Problems	2–4	Moderate–High

8.0Most Common Question Types

Finding the (n^{th}) term of an Arithmetic Progression.
Calculating the sum of the first (n) terms of an AP.
Finding Arithmetic Mean(s) between given numbers.
Solving Geometric Progression problems.
Finding the sum of finite and infinite GPs.
Determining Geometric Mean(s) between two numbers.
Solving mixed AP-GP application-based questions.
JEE Advanced problems involving progression-based reasoning.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise the concepts of sequences and series.
Memorize all AP formulas and practice finding terms.
Revise the sum formulas for Arithmetic Progressions.
Learn GP formulas and common ratio concepts.
Practice finite and infinite GP problems.
Revise Arithmetic Mean and Geometric Mean formulas.
Solve Previous Years' Questions (PYQs).
Finish with mixed practice sets involving AP, GP, and application-based questions.

Test your Knowledge

question 1 of 5

The common difference of the AP 3, 7, 11, 15, ... is:

Frequently Asked Questions

How can I prepare Sequences and Series effectively for JEE?

Are ALLEN's Class 11 Maths Chapter 8 Sequences and Series Revision Notes sufficient for quick revision?

What is Class 11 Maths Chapter 8: Sequences and Series about?

Why are Sequences and Series important for JEE Main and JEE Advanced?

Which topics should I focus on while revising Sequences and Series?

Join ALLEN!

Class 11 Maths Chapter 8 Sequences and Series Revision Notes

1.0Chapter Snapshot

2.0Chapter 8 Smart Study Material by ALLEN Experts: Download PDFs

Test your Knowledge

question 1 of 5

The common difference of the AP 3, 7, 11, 15, ... is:

Sequence and Series

1.0Sequence and Series Definition

2.0Types of Sequence and Series

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

Frequently Asked Questions

How can I prepare Sequences and Series effectively for JEE?

Are ALLEN's Class 11 Maths Chapter 8 Sequences and Series Revision Notes sufficient for quick revision?

What is Class 11 Maths Chapter 8: Sequences and Series about?

Why are Sequences and Series important for JEE Main and JEE Advanced?

Which topics should I focus on while revising Sequences and Series?

Join ALLEN!

Class 11 Maths Chapter 8 Sequences and Series Revision Notes

1.0Chapter Snapshot

2.0Chapter 8 Smart Study Material by ALLEN Experts: Download PDFs

Sequence and Series

1.0Sequence and Series Definition

2.0Types of Sequence and Series

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy