RD Sharma Class 11 Solutions Chapter 12 - Mathematical Induction (Ex 12.1) Exercise 12.1 - Free PDF
The PDF here contains the solutions for Chapter 12 - Mathematical Induction of RD Sharma Class 11. Students who want to score good marks in exams and want to prepare for the topic ‘mathematical induction’ thoroughly can refer to this PDF. It has precise information and solutions developed by the subject matter experts to give students a complete overview of this chapter and to assist them in solving all the problems in the exercise.
The technique called mathematical induction is used to prove a theorem, a statement, or a formula that is true for every natural number. Mathematical induction is one of those techniques which can be used to prove a variety of mathematical statements which are formulated in terms of n, where n belongs to the set of positive integers. There are two steps to solve problems of mathematical induction.
Base or initial step: It gives proof that the statement is true for an initial value.
Inductive step: It proves that when a statement is true for an iteration, say nth , then it also holds true for the iteration (n+1)th.
More generally, we can also use the mathematical induction technique to prove that a propositional function, say P(n), is true for all the integers n≥1.
A natural beginning point for proving many mathematical results is to look at a few simple cases like if a statement is true for an initial value. This helps us get some pointers for finding proof.
We are trying to prove that not one proposition is true, but a whole sequence of propositions are true, one for each n. The trick used in mathematical induction is to prove the given mathematical statement in the sequence, and then prove that if any particular statement is true, then the one after it is also true. This enables us to conclude that all the statements are true in a sequence.
1. Consider if P (n) is a statement “n (n + 1) is even”, then what is P (3)?
P (n) = n (n + 1) is even.
P (3) = 3 (3 + 1)
= 3 (4)
So, P (3) = 12, P (3) is also even.
2. Consider if P (n) is a statement “n3 + n is divisible by 3”, then prove that P (3) is true, but P (4) is not true.
Now, P (n) = n3 + n is divisible by 3
So We have P (n) = n3 + n
P (3) = 33 + 3
= 27 + 3
P (3) = 30, Hence it is divisible by 3
Now, let’s check with P (4)
P (4) = 43 + 4
= 64 + 4
P (4) = 68, so it is not divisible by 3
So, P (3) is true and P (4) is not true.
As mentioned earlier, mathematical induction is an easy topic in the Class 11 syllabus when you have the right study material to follow. This is why you need the solution PDF file for all the exercises in this chapter to study and to understand the specific approaches defined by the subject matter experts of Vedantu. Download it today and add it to your study schedule to prepare this chapter better.
FAQs on Free Download of RD Sharma Solutions PDF for Class 11 Maths Chapter 12 For Best Study Support
1. Is it necessary to solve and learn every single question from the RD Sharma class 11 solutions for chapter 12 PDF?
Yes. It is necessary that students must learn, practice, and solve every question from the RD Sharma class 11 solutions for chapter 12 PDF as these questions may appear in any kind of exam and it has been made so that the students can have a quick learning session with all the necessary information that can help them score good marks in their exams. This PDF is a performance booster for students when it comes to scoring good marks in the board exams, entrance exams, etc. To solve these questions more accurately in exams, students need good practice.
2. What are the benefits of practicing RD Sharma Class 11 Solutions Chapter 12?
Practicing RD Sharma Class 11 Solutions Chapter 12 can give students a clear idea about the topic of mathematical induction and helps students gain an experience of the sample questions that can appear in board exams, entrance exams of engineering, etc. It contains all the important information required to get a good hold of the subject. These solutions cover all the necessary information that is included in the recent and updated syllabus.
3. Can students expect a question from mathematical induction in any exams?
Mathematical induction is one the most unique topics from which questions can be asked in any kind of exam. Questions can be framed ranging from easy to hard and the theorems and steps of mathematical induction can be applied while solving other problems as well. So, preparing mathematical induction is necessary even if the question may or may not be asked in exams.
4. Can Vedantu’s RD Sharma class 11 solutions for chapter 12 help score the best marks in exams?
Yes. The RD Sharma class 11 solutions for chapter 9 PDF is a good source of practice for the exams. Students will get to know what all types of questions can be framed and what logical approach they need to apply to the problem to solve it. For board exams, this PDF might be enough but considering the unpredictable pattern of the difficulty level of the board exam question papers, it is always advised that students also prepare from multiple reference books so that they can score better. It also will in turn be of good help for them in further studies.
5. How necessary is it to solve and learn every single question from the RD Sharma class 11 solutions for chapter 9 PDF?
Students must learn, practice, and solve every question from the RD Sharma class 11 solutions for chapter 9 PDF as these questions may appear in any kind of exam and it has been made so that the students can have a quick and fun learning session with all the necessary information that can help them score good marks in their exams. This PDF is an academic performance booster for students when it comes to scoring in the board exams, entrance exams, etc. To solve these questions more accurately in exams, students need good practice of these questions to solve them well in the exams eventually.