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RD Sharma Solutions

RD Sharma Class 11 Maths Solutions Chapter 19 - Arithmetic Progressions

RD Sharma Class 11 Maths Solutions Chapter 19 - Arithmetic Progressions

Q: 5. Why is it strategically better to select three consecutive terms in an AP as (a-d), a, and (a+d) for solving certain problems?

Choosing terms as (a-d), a, (a+d) is a powerful problem-solving strategy, especially when the sum of these terms is given. When you add these terms, the common difference 'd' cancels out: (a-d) + a + (a+d) = 3a. This immediately gives you the value of 'a' (the middle term), simplifying the problem significantly. This technique reduces the number of variables you need to solve for simultaneously, making the solution quicker and less prone to errors compared to using a, a+d, and a+2d.

Q: 6. How can you definitively verify if a given sequence is an Arithmetic Progression when solving a problem?

The fundamental property of an AP is its constant common difference. To verify if a sequence is an AP, you must check if the difference between any two consecutive terms is the same throughout the sequence. For a sequence a₁, a₂, a₃, ..., aₙ, you must confirm that a₂ - a₁ = a₃ - a₂ = a₄ - a₃, and so on. If even one pair of consecutive terms does not yield the same difference, the sequence is not an Arithmetic Progression.

RD Sharma Solutions for Class 11 Maths Chapter 19 - Free PDF Download

Arithmetic Progression is a number series in which a constant value is the difference between any two consecutive numbers. In real life, Arithmetic Progression is important because this helps us to understand things through the use of patterns. RD Sharma Class 11 Arithmetic Progression explains important concerts of Arithmetic Progression and provides solutions to some of the most important and frequently asked questions in board exams. Vedantu provides a free PDF of RD Sharma Class 11 Chapter 19 Solutions.

Students can download these free PDFto understand the Arithmetic Progression concepts from the top faculty and score good marks in their exams. The RD Sharma Class 11 Ch 19 Solutions are prepared according to the NCERT curriculum. The solutions are prepared in a step by step manner by explaining all steps and important formulas so that the students will easily understand the concepts.

RD Sharma Class 11 Arithmetic Progression

The RD Sharma Solutions For Class 11 Maths Chapter 19 also has a lot of exercises and practice problems for students so that it will improve their confidence level before appearing for the exams.

Chapter 19 of the Class 11 Maths syllabus is a very essential chapter for the students. The chapter deals with the relevant basics which should be known to each student. These basics will help them understand other related chapters of the subject as well. The fundamentals of these chapters should be thoroughly known to each student. Also, the questions related to this chapter can be solved easily and quickly by the students during their exams. Moreover, questions from the concepts related to this chapter are also asked in competitive exams and general aptitude tests.

Arithmetic progression generally refers to a mathematical sequence wherein the difference between the two consecutive terms is always constant. Arithmetic progression is generally written as AP in its abbreviated form. An arithmetic progression is also referred to as a sequence of numbers where for every consecutive term, the second number is obtained by adding a specific number to the first one. The fixed number that is added to the first number to get the second number is called the common difference of an arithmetic sequence.

Here let us look at a few of the important topics from the Arithmetic Progression chapter which are frequently asked in exams.

Definition of a Sequence.
Definition of an Arithmetic Progression.
The general term of an Arithmetic Progression.
Selection of terms in an Arithmetic Progression.
Sum to n terms of an Arithmetic Progression.
Properties of Arithmetic Progression.
Insertion of arithmetic means.
Applications of Arithmetic Progression.

Exercises in RD Sharma Solutions for Class 11 Maths Chapter 19

Exercise-wise Description of RD Sharma Class 11 Chapter 19

RD Sharma Class 11 Chapter 19 contains a total of seven exercises.

The first exercise, that is, 19.1 of RD Sharma Class 11 Maths Chapter 19 contains basic questions wherein students will be required to find terms on the basis of a given nth term or a particular sequence.
The second exercise, that is, 19.2 of RD Sharma Class 11 Chapter 19 will enable students to find the terms on the basis of the given A.P. in the first four questions of the exercise. The difficulty level of the rest of the questions of the exercise keeps on increasing.
Exercises 19.3 -- 19.7 of RD Sharma chapter 19 are more complex and contain more difficult questions. There are a total of 57 questions in these five exercises.

List of Formulas

There are certain formulae present in class 11 chapter 19, that is, an arithmetic progression which is very essential for the students to remember:

The general form of an arithmetic progression is given by a, a+d, a+2d, a+3d,....
The nth term of an arithmetic progression is given as an = a + (n – 1) × d
Sum of n terms in an arithmetic progression is given by: S = n/2 [2a + (n - 1) x d
Sum of all terms in a finite arithmetic progression with the last term as ‘I’ will be given as n/2 (a+l)
d is a common difference and S is the sum of the first n terms, When n is the nth term of the sequence,

Tips to Prepare for Exams using RD Sharma Solutions For Class 11 Maths Chapter 19

These tips on the Arithmetic Progression chapter will help students to score good marks in their exams.

First, brush up on the basic concepts of sequence and series and types of sequences asked in exams.
All the important questions from Arithmetic Progression deal with direct formula based solutions. So it is important to remember all the formulas used such as the general term of an Arithmetic Progression and the sum of Arithmetic Progression.
Read the questions carefully when asked from sequence and series because we have 3 types of series so depending on the given question apply the formulas of Arithmetic Progression and solve accordingly.

Conclusion

The RD Sharma Class 11 Ch 19 Solutions prepared by Vedantu is helpful for students to score good marks in their board exams. These solutions are prepared based on important questions from the NCERT curriculum by the top faculty of Vedantu. The practice problems provided with Arithmetic Progression RD Sharma Solutions will help students to revise the concepts and ace their exams. The RD Sharma Solutions on Arithmetic Progression are prepared by experts to provide top-notch learning content to students. Experts have done a lot of research on the preparation of solutions to provide a unique and fun learning experience to students.

FAQs on RD Sharma Class 11 Maths Solutions Chapter 19 - Arithmetic Progressions

1. What are the major topics a student will encounter in the RD Sharma Class 11 Maths Solutions for Chapter 19, Arithmetic Progressions?

The solutions for this chapter systematically cover all essential concepts of Arithmetic Progressions (AP) as per the Class 11 syllabus. The key areas include:

Definition and identification of a sequence as an AP.
Finding the general term (nth term) of an AP.
Methods for selecting terms in an AP for problem-solving.
Calculating the sum of the first n terms of an AP.
Understanding and applying the properties of Arithmetic Progressions.
The procedure for inserting a specified number of Arithmetic Means (AMs) between two given numbers.

2. What is the standard step-by-step method to find the nth term (aₙ) of an AP, as demonstrated in RD Sharma solutions?

To find the nth term of an Arithmetic Progression, the solutions consistently apply the standard formula aₙ = a + (n-1)d. The correct method involves these steps:

Identify the first term (a) of the sequence.
Calculate the common difference (d) by subtracting any term from its succeeding term (e.g., a₂ - a₁).
Determine the position of the term you need to find, which is 'n'.
Substitute the values of 'a', 'd', and 'n' into the formula to compute the value of aₙ.

3. How do the RD Sharma solutions explain the calculation for the sum of the first 'n' terms (Sₙ) of an AP?

The solutions illustrate two primary formulas for finding the sum of 'n' terms. The choice of formula depends on the information given in the problem:

When the last term is unknown: Use the formula Sₙ = n/2 [2a + (n-1)d]. You need the first term (a), the common difference (d), and the number of terms (n).
When the last term (l) is known: A more direct formula is used, Sₙ = n/2 [a + l]. This is efficient as it only requires the first term, the last term, and the number of terms.

The solutions guide you to first identify the known variables and then select the appropriate formula for an accurate calculation.

4. How does one solve problems on inserting multiple Arithmetic Means (AMs) between two numbers as per the methods in RD Sharma?

To insert 'n' Arithmetic Means between two numbers, say 'a' and 'b', the method involves creating a new AP. Here are the steps:

Let the 'n' AMs be A₁, A₂, ..., Aₙ. The new sequence will be a, A₁, A₂, ..., Aₙ, b.
This new sequence is an AP with a total of (n+2) terms. Here, the first term is 'a' and the (n+2)th term is 'b'.
Use the nth term formula to find the common difference (d). Set up the equation: b = a + ((n+2) - 1)d.
Solve this equation for 'd'.
Once 'd' is found, the AMs can be calculated as: A₁ = a + d, A₂ = a + 2d, and so on, up to Aₙ = a + nd.

5. Why is it strategically better to select three consecutive terms in an AP as (a-d), a, and (a+d) for solving certain problems?

Choosing terms as (a-d), a, (a+d) is a powerful problem-solving strategy, especially when the sum of these terms is given. When you add these terms, the common difference 'd' cancels out: (a-d) + a + (a+d) = 3a. This immediately gives you the value of 'a' (the middle term), simplifying the problem significantly. This technique reduces the number of variables you need to solve for simultaneously, making the solution quicker and less prone to errors compared to using a, a+d, and a+2d.

6. How can you definitively verify if a given sequence is an Arithmetic Progression when solving a problem?

The fundamental property of an AP is its constant common difference. To verify if a sequence is an AP, you must check if the difference between any two consecutive terms is the same throughout the sequence. For a sequence a₁, a₂, a₃, ..., aₙ, you must confirm that a₂ - a₁ = a₃ - a₂ = a₄ - a₃, and so on. If even one pair of consecutive terms does not yield the same difference, the sequence is not an Arithmetic Progression.

7. What is a common conceptual mistake to avoid when applying the properties of an AP to solve complex questions?

A common mistake is incorrectly applying a property without understanding its conditions. For instance, while it's true that if you add, subtract, multiply, or divide each term of an AP by a non-zero constant, the resulting sequence is also an AP, students often miscalculate the new common difference. If the original common difference is 'd' and you multiply each term by 'k', the new common difference becomes 'kd', not 'd'. Forgetting to adjust the common difference can lead to completely wrong answers in subsequent steps.

8. When solving word problems from RD Sharma, how do you translate the problem's language into the mathematical components (a, d, n) of an AP?

Translating word problems requires careful identification of key phrases:

The "initial amount," "starting value," or "first instalment" usually represents the first term (a).
A "constant increase/decrease," "uniform change," or "annual increment" typically signifies the common difference (d). Remember that a decrease implies a negative 'd'.
The "number of years," "number of terms," or "time period" corresponds to 'n'.
A question asking for a value at a "specific time" (e.g., "in the 10th year") requires you to find the nth term (aₙ), whereas a question about the "total amount" or "cumulative value" requires the sum of n terms (Sₙ).