NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series (Ex 9.4) Exercise 9.4

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Mathematics is one of the most challenging subjects for some students. Class 11 has a completely different concept of mathematics from Class 10. Moreover, mathematics is one of the subjects used in almost every career field, and many career options are available to students after studying Mathematics in Class 11. It is very important to understand the formulas and concepts in Mathematics to clear up any doubts about the basics.

In Class 11, students study for various health, technology, law, and management entrance examinations. These entrance tests make use of concepts learned in Class 11. The entrance examination has a very vast syllabus, and mostly it is a basic syllabus for Classes 11 and 12. Moreover, school students undervalue Class 11 because it appears to be the basis for advancing to better academic tiers, such as Class 12. However, it is essential to think of Class 11 as equally significant for finishing their school education.

Class 11 topics are comprehensive, necessitating regular revision and practice. Furthermore, exercise questions can be difficult for learners to answer immediately. Extramarks provides a variety of study materials, such as NCERT solutions, PDF notes, and other materials, to help students thoroughly prepare for each section of their examinations. The NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4, can help students enhance their exam preparation. The NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4, are based on the most up-to-date CBSE syllabus. Class 11 Mathematics Chapter 9 Exercise 9.4 NCERT Solutions are necessary for answering Class 11 Maths Chapter 9 Exercise 9.4 questions.

NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4, are available as PDF files for Class 11 Mathematics students. As a result, using NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4, students study from the Extramarks website and mobile application, which is a trustworthy and reliable source of information. Extramarks makes it simpler for students to access NCERT Solutions Class 12 for all subjects. NCERT Solutions Class 11 for all subjects are also easily accessible to students in Class 11. Additionally, students seeking NCERT Solutions Class 10 in English can usually find them on the Extramarks website and mobile app too. Mathematics NCERT Solutions Class 9 are also available on Extramarks for each exercise at the end of every chapter.

On the other hand, students in Class 8 could access NCERT Solutions Class 8 for all topics. Furthermore, NCERT Solutions Class 7 for all Mathematics sections are available on the Extramarks website and mobile app as a PDF download. Students in Class 6 must utilize the NCERT Solutions Class 6. NCERT Solutions Class 5 are also available on Extramarks for all subjects. Students in Class 4 can begin preparing for their exams by downloading the NCERT Solutions Class 4. Moreover, NCERT Solutions Class 3 are now available on the Extramarks website and mobile app. NCERT Solutions Class 2 are available at Extramarks for all exercises in all subjects. Class 1 students can access NCERT Solutions Class 1 for all subject as well.

NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series (Ex 9.4) Exercise 9.4

In Class 11 mathematics Chapter 9, arithmetic progression and geometric progression are discussed, as well as their correlation. In order to understand these topics well, students should practice the NCERT solutions. Chapter 9 of Class 11 is quite challenging. Some schoolchildren may be unsure of how to solve the practice questions. A good practice set of questions for Chapter 9 can be found in the NCERT books. Students can use the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4 to assist them in completing the exercise. Extramarks offers live sessions on their website and mobile app where students can clarify their doubts about the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4.

Students who are having difficulty with Chapter 9 can refer to the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4. Students can simply get the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4 and start solving their answers on the Extramarks website and mobile application.

What You’ll Learn in Exercise 9.4 of Class 11 Maths Chapter 9 Sequences and Series?

The NCERT Solutions for Class 11 Mathematics Chapter 9- Sequences and Series Exercise 9.4, explains how and where to determine the sum of the ‘n’ terms of a special series. There are various types of questions based on consecutive numbers in Chapter 9 Sequences and Series (Ex 9.4), such as the sum of consecutive squares, the sum of consecutive cubes, or the sum of consecutive natural numbers. In Chapter 9 of Class 11 Mathematics, students will learn about Arithmetic Progression and Geometric Progression, as well as interesting topics associated with them. Students should focus on practicing NCERT solutions to understand these topics thoroughly. Students should always do the exercises on a daily practice basis. Students can understand the exercise by using the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4.

Access NCERT Solutions for Class 11 Maths Chapter 9 – Sequences and Series

The Extramarks website and mobile application make it simple to find NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4. Using the best study materials when students start studying for the exam streamlines the process. There is a great deal of stress in Class 11, and students are confused about how to manage all of their subjects for exams.

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NCERT Solutions for Class 11 Maths Chapters

The practice of exercise questions is another important aspect of scoring well in Class 11 Mathematics, apart from revising the concepts.

Chapter 1: Sets

Sets are a crucial chapter in Mathematics, used to classify the principles of relations and capacities. It will cover sets in depth, focusing on the set hypothesis, finite and infinite sets, and so on.

Chapter 2: Relations and Functions

Students will learn about the Mathematical correspondence between quantities and the relationships between objects, functions, and functions.

Chapter 3: Trigonometric Functions

In Chapter 3, students will be guided through more advanced trigonometric concepts, such as trigonometric ratios and functions. Students will also study their characteristics.

Chapter 4: Principle of Mathematical Induction

The chapter on Mathematical Induction is very interesting. A fundamental concept of Mathematical Induction is discussed in this section. Students will then be expected to apply their ethics to various situations.

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 5 will assist students in comprehending complex numbers and solving problems based on them. It will also cover negative number square roots, the Argand Plane, and quadratic equations.

Chapter 6: Linear Inequalities

In this section, a diverse range of applications of linear inequalities are studied. These include algebraic solutions to linear inequalities in one variable, graphical solutions to linear inequalities in two variables, and graphical solutions to linear inequalities in three variables.

Chapter 7: Permutations and Combinations

Permutations, combinations, permutations with distinct or non-distinct objects, factorial notation, and other related topics will be covered in this chapter.

Chapter 8: Binomial Theorem

Chapter 8 presents the binomial theorem for positive integral indices, Pascal’s triangle, and a wide range of applications of the binomial theorem.

Chapter 9: Sequences and Series

Chapter ‘Sequences and Series’ introduces students to Arithmetic Progression, Geometric Progression, General Terms of General Progressions, and sums to ‘n’ terms of General Progressions. It is a required component of Class 11 Mathematics. The exercise in Chapter 9 is fascinating. Moreover, students facing a few issues solving Math Class 11 Chapter 9 Exercise 9.4 can find help from the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4.

Chapter 10: Straight Lines

This chapter introduced students to a variety of important straight-line concepts. It will be divided into topics such as line slope, the angle between two lines, collinearity between two points, and many more.

Chapter 11: Conic Sections

In Chapter 11, these concepts will be expanded, and students will be introduced to conic segments such as parabolas and hyperbolas.

Chapter 12: Introduction to Three Dimensional Geometry

In this chapter, students will find out about three-dimensional coordinate geometry.

Chapter 13: Limits and Derivatives

Students will learn about derivatives and limits and the limits and derivatives of polynomials and trigonometric functions.

Chapter 14: Mathematical Reasoning

Statements, Statement Negation, Compound Statements, Quantifiers, Implications, Validating Statements, and many other concepts will be covered in Chapter 14.

Chapter 15: Statistics

This chapter will expand on those fundamentals by discussing dispersion measures, range, mean deviation, standard deviation, and so on.

Chapter 16: Probability

Students will learn more about events, types of events, mutually exclusive events, specific events, and probabilities of different events in Chapter 16 of the Class 11 Mathematics NCERT book.

NCERT Solution Class 11 Maths of Chapter 9 Exercises

Class 11 Chapter 9 Mathematics is challenging for the students. Practicing is very important to understand the formulas. This chapter contains a good number of exercises. This helps students clear their doubts about this chapter. Moreover, many students may be facing some challenges while solving the practice. In that case, Extramarks is one of the best platforms for having the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4.

Students can solve and practise a vast number of questions in the NCERT textbook. Starting to practice with the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4, will help students achieve good marks in their Class 11 exams. Students having difficulties with Chapter 9 should refer to the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4. Students can easily access the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4, on the Extramarks website and mobile app.

NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series Exercise 9.4

Chapter 9 of Class 11 Mathematics is tricky for some students, and it takes a lot of practice to understand the formulas. When students start preparing for their exams, they have a lot of pressure from other subjects. Now, students want the best solutions to their Mathematical problems. Extramarks is one of the most amazing learning platforms for students to refer to in order to solve their problems. Students can attend live lectures and interact with the teacher to solve their queries about the topic.

For example, if students have difficulties understanding Class 11th Maths Chapter 9 Exercise 9.4, they can access the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4 on the Extramarks website and mobile application. Even after practicing questions from Class 11th Math 9.4, students can check their answers with the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4.

There are several exercises in Chapter 9, but the fourth one is a bit difficult. Many students become puzzled while attempting to solve Class 11th Math 9.4. Students can supplement their practice with the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4. Students can find the NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.4, on the Extramarks website and mobile app.

Q.1 Find the sum to n terms of each of the series in Exercises 1 to 7.

1. 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 +…
2. 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + …
3. 3 × 1² + 5 × 2² + 7 × 3² + …
4.

$\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + …$

5. 5² + 6² + 7² + … + 20²
^6.^{3 × 8 + 6 × 11 + 9 × 14 + …}
^7.^{1² + (1² + 2²) + (1² + 2² + 3²) + …}

Ans.

$\begin{array}{l} 1 . Let S_{n} = 1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + . .. + n \times (n + 1) \\ a_{n} = n \times (n + 1) \\ S_{n} = \sum n \times (n + 1) \\ = \sum n^{2} + \sum n \\ = \frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2} \\ = \frac{n (n + 1)}{2} (\frac{2 n + 1}{3} + 1) \\ = \frac{n (n + 1)}{2} (\frac{2 n + 4}{3}) \\ = \frac{n (n + 1)}{2} \times 2 (\frac{n + 2}{3}) \\ S_{n} = \frac{1}{3} n (n + 1) (n + 2) \\ 2. Let S_{n} = 1 \times 2 \times 3 + 2 \times 3 \times 4 + 3 \times 4 \times 5 + . .. + n \times (n + 1) \times (n + 2) \\ a_{n} = n \times (n + 1) \times (n + 2) \\ S_{n} = \sum n \times (n + 1) \times (n + 2) \\ = \sum (n^{3} + 3 n^{2} + 2 n) \\ = \sum n^{3} + 3 \sum n^{2} + 2 \sum n \\ = {\{\frac{n (n + 1)}{2}\}}^{2} + 3 . \frac{n (n + 1) (2 n + 1)}{6} + 2 . \frac{n (n + 1)}{2} \\ = \frac{n^{2} {(n + 1)}^{2}}{4} + \frac{n (n + 1) (2 n + 1)}{2} + \frac{2 n (n + 1)}{2} \\ = \frac{n (n + 1)}{2} \{\frac{n (n + 1)}{2} + (2 n + 1) + 2\} \\ = \frac{n (n + 1)}{2} \{\frac{n^{2} + n + 4 n + 2 + 4}{2}\} \\ = \frac{n (n + 1)}{2} (\frac{n^{2} + 5 n + 6}{2}) \\ = \frac{n (n + 1) (n + 2) (n + 3)}{4} \\ S_{n} = \frac{1}{4} n (n + 1) (n + 2) (n + 3) \\ 3 . a_{n} = (2 n + 1) \times n^{2} \\ S_{n} = \sum (2 n^{3} + n^{2}) \\ = 2 \sum n^{3} + \sum n^{2} \\ = 2 {\{\frac{n (n + 1)}{2}\}}^{2} + \frac{n (n + 1) (2 n + 1)}{6} \\ = 2 . \frac{n^{2} {(n + 1)}^{2}}{4} + \frac{n (n + 1) (2 n + 1)}{6} \\ = \frac{n^{2} {(n + 1)}^{2}}{2} + \frac{n (n + 1) (2 n + 1)}{6} \\ = \frac{n (n + 1)}{2} \{n (n + 1) + \frac{(2 n + 1)}{3}\} \\ = \frac{n (n + 1)}{2} \{\frac{3 n (n + 1) + (2 n + 1)}{3}\} \\ = \frac{n (n + 1)}{2} \{\frac{3 n^{2} + 3 n + 2 n + 1}{3}\} \\ = \frac{n (n + 1)}{2} \{\frac{3 n^{2} + 5 n + 1}{3}\} \\ S_{n} = \frac{1}{6} n (n + 1) (3 n^{2} + 5 n + 1) \\ 4 . Let S_{n} = \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + . .. + \frac{1}{n (n + 1)} \\ = (\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + . .. + (\frac{1}{n} - \frac{1}{n + 1}) \\ = \frac{1}{1} - \frac{1}{n + 1} \\ = \frac{n + 1 - 1}{n + 1} \\ = \frac{n}{n + 1} \\ 5. 5^{2} + 6^{2} + 7^{2} + . .. + 20^{2} \\ = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + 7^{2} + . .. + 20^{2} - (1^{2} + 2^{2} + 3^{2} + 4^{2}) \\ = \sum 20^{2} - \sum 4^{2} \\ = \frac{20 (20 + 1) (2 \times 20 + 1)}{6} - \frac{4 (4 + 1) (2 \times 4 + 1)}{6} \\ = \frac{20 \times 21 \times 41}{6} - \frac{4 \times 5 \times 9}{6} \\ = \frac{17220 - 180}{6} \\ = \frac{17040}{6} \\ = 2840 \\ 6 . {Let S}_{n} = 3 \times 8 + 6 \times 11 + 9 \times 14 + . .. \\ T_{n} = (3, 6, 9, . .. n^{th} term) (8, 11, 14, . .. n^{th} term) \\ = \{3 + (n - 1) 3\} \{8 + (n - 1) 3\} \\ = (3 n) (3 n + 5) \\ = 9 n^{2} + 15 n \\ S_{n} = \sum T_{n} \\ = \sum (9 n^{2} + 15 n) \\ = 9 \sum n^{2} + 15 \sum n \\ = 9 \times \frac{n (n + 1) (2 n + 1)}{6} + 15 \times \frac{n (n + 1)}{2} \\ = 3 \times \frac{n (n + 1) (2 n + 1)}{2} + 15 \times \frac{n (n + 1)}{2} \\ = \frac{3 n (n + 1)}{2} \{(2 n + 1) + 5\} \\ = \frac{3 n (n + 1)}{2} (2 n + 6) \\ = 3 n (n + 1) (n + 3) \\ 7 . Let S_{n} = 1^{2} + (1^{2} + 2^{2}) + (1^{2} + 2^{2} + 3^{2}) + . .. + (1^{2} + 2^{2} + 3^{2} + . .. + n^{2}) \\ Here, T_{n} = 1^{2} + 2^{2} + 3^{2} + . .. + n^{2} \\ = \sum n^{2} \\ = \frac{n (n + 1) (2 n + 1)}{6} \\ = \frac{1}{6} n (2 n^{2} + 3 n + 1) \\ = \frac{1}{6} (2 n^{3} + 3 n^{2} + n) \\ S_{n} = \sum T_{n} \\ = \frac{1}{6} \sum (2 n^{3} + 3 n^{2} + n) \\ = \frac{1}{6} (2 \sum n^{3} + 3 \sum n^{2} + \sum n) \\ = \frac{1}{6} [2 {\{\frac{n (n + 1)}{2}\}}^{2} + 3 \frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2}] \\ = \frac{1}{6} \{\frac{n^{2} {(n + 1)}^{2}}{2} + \frac{n (n + 1) (2 n + 1)}{2} + \frac{n (n + 1)}{2}\} \\ = \frac{1}{6} \times \frac{n (n + 1)}{2} \{n (n + 1) + (2 n + 1) + 1\} \\ = \frac{1}{6} \times \frac{n (n + 1)}{2} (n^{2} + n + 2 n + 2) \\ = \frac{1}{12} n (n + 1) (n^{2} + 3 n + 2) \\ = \frac{1}{12} n (n + 1) (n + 1) (n + 2) \\ S_{n} = \frac{n {(n + 1)}^{2} (n + 2)}{12} \end{array}$

Q.2 Find the sum to n terms of the series in whose n^th terms is given by:

1. n (n + 1) (n + 4).
2. n² + 2ⁿ
^3.^{(2n – 1)²}

Ans.
1.

${Let T}_{n} = n (n + 1) (n + 4)$ $\begin{array}{l} = n (n^{2} + 5 n + 4) \\ = n^{3} + 5 n^{2} + 4 n \\ S_{n} = \sum T_{n} \\ = \sum (n^{3} + 5 n^{2} + 4 n) \\ = \sum n^{3} + 5 \sum n^{2} + 4 \sum n \\ = {\frac{n (n + 1)}{2}}^{2} + \frac{5 n (n + 1) (2 n + 1)}{6} + \frac{4 n (n + 1)}{2} \\ = \frac{n (n + 1)}{2} {\frac{n (n + 1)}{2} + 5 . \frac{(2 n + 1)}{3} + 4} \\ = \frac{n (n + 1)}{2} {\frac{3 n (n + 1) + 10 (2 n + 1) + 24}{6}} \\ = \frac{n (n + 1)}{2} (\frac{3 n^{2} + 3 n + 20 n + 10 + 24}{6}) \\ S_{n} = \frac{n (n + 1)}{12} (3 n^{2} + 23 n + 34) \end{array}$

$\begin{array}{l} Let T_{n} = n^{2} + 2^{n} \\ S_{n} = \sum T_{n} \\ = \sum (n^{2} + 2^{n}) \\ = \sum n^{2} + \sum 2^{n} \\ = \frac{n (n + 1) (2 n + 1)}{6} + (2 + 2^{2} + 2^{3} + 2^{4} + . .. + 2^{n}) \end{array}$ $\begin{array}{l} \end{array}$ $\begin{array}{l} \end{array}$

$\begin{array}{l} = \frac{n (n + 1) (2 n + 1)}{6} + \frac{2 (2^{n} - 1)}{2 - 1} [∵ S_{n} = \frac{a (r^{n} - 1)}{r - 1}] \\ = \frac{n (n + 1) (2 n + 1)}{6} + 2 (2^{n} - 1) \\ ∴ S_{n} = \frac{1}{6} n (n + 1) (2 n + 1) + 2 (2^{n} - 1) \end{array}$

$\begin{array}{l} Let t_{n} = {(2n - 1)}^{2} \\ = 4 n^{2} - 4 n + 1 \\ S_{n} = \sum (4 n^{2} - 4 n + 1) \\ = 4 \sum n^{2} - 4 \sum n + \sum 1 \\ = 4 \frac{n (n + 1) (2 n + 1)}{6} - 4 \frac{n (n + 1)}{2} + n \\ = 2 \frac{n (n + 1) (2 n + 1)}{3} - 2 n (n + 1) + n \\ = \frac{2 n (2 n^{2} + 3 n + 1)}{3} - 2 n^{2} - 2 n + n \\ = \frac{2 (2 n^{3} + 3 n^{2} + n)}{3} - 2 n^{2} - n \\ = \frac{4 n^{3} + 6 n^{2} + 2 n - 6 n^{2} - 3 n}{3} \\ = \frac{4 n^{3} - n}{3} \end{array}$ $\begin{array}{l} \end{array}$

$\begin{array}{l} = \frac{n}{3} (4 n^{2} - 1) \\ S_{n} = \frac{n}{3} (2 n - 1) (2 n + 1) \end{array}$

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NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series (Ex 9.4) Exercise 9.4

NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series (Ex 9.4) Exercise 9.4

What You’ll Learn in Exercise 9.4 of Class 11 Maths Chapter 9 Sequences and Series?

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NCERT Solutions for Class 11 Maths Chapters

Chapter 1: Sets

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

NCERT Solution Class 11 Maths of Chapter 9 Exercises

NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series Exercise 9.4

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NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series (Ex 9.4) Exercise 9.4

NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series (Ex 9.4) Exercise 9.4

What You’ll Learn in Exercise 9.4 of Class 11 Maths Chapter 9 Sequences and Series?

Access NCERT Solutions for Class 11 Maths Chapter 9 – Sequences and Series

NCERT Solutions for Class 11 Maths Chapters

Chapter 1: Sets

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

NCERT Solution Class 11 Maths of Chapter 9 Exercises

NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series Exercise 9.4

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