RD Sharma Class 11 Chapter 12 Solutions (Mathematical Induction)
RD Sharma Solutions for Class 11 Chapter 12 help you understand the chapter completely. These solutions will help you learn about mathematical statements, mathematical induction, its history, principles of Mathematical Induction, properties of both the Principles, the difference between first and second principle and Inductive Hypothesis. All these topics are important for CBSE exam purposes.
In RD Sharma Class 11 Solutions for Maths Chapter 12, we will solve two exercises containing 57 Questions of proving mathematical statements by applying principles of mathematical induction. You will need to put assumptions in various steps to get the right solution. Identifying the right principle to use is not that easy but once the concept gets clear, everything falls at the right place. Few multiple questions will demand to try each option to hit the true one.
Instasolv is here to let you comprehend with the exact touch of mathematical induction. As this chapter is full of assumptions, solving every question with the same approach might not work. Our RD Sharma Class 11 Chapter 12 Solutions will help you to understand various methods of solving the questions and also allow you to identify either of the principles while solving different mathematical statements.
Topics covered in RD Sharma class 11 chapter 12
Mathematical induction & its History
An implied hint of mathematical induction was found when Euclid tried to prove the infinite prime numbers. Later it was also seen in the cyclical method of Bhaskara.
Mathematical Induction is a technique used to prove statements and theorems are correct for every set of natural numbers. The simple idea is by repeated succession, one can reach any natural number. Mathematical induction comprises two steps. The foremost that is the base step says that the statement holds true for the first natural number. The second step which is the induction step confirms that the statement is true for all the natural numbers. Here, the notation used with a natural number (n) is P.
As we witnessed that mathematical induction is a game of proving desired statements. For this, one needs to assume ‘n’ in many cases. More often, the assumption is required to prove that the statement holds true for n +1. Also, to prove the statement of P(k+1) we need to assume P(k). This assuming step is called an inductive hypothesis.
Principle of Mathematical Induction
Mathematical induction works with two principles.
- First principle: It requires three steps to conclusively prove the statement. The first principle aims to prove that if P(k) is true, then P(k+1) is also true.
- Second principle: The principle aims to prove the statement that if n= r+1 then, m=n, n+1, n+2… r (where m is a fixed integer). It is said that the second principle is stronger than the first.
Here is a guide to prove mathematical statements mainly comprises three subsequent steps are
- Initially, you have to Let n=1.
- Next step is an inductive hypothesis where you will have to assume P(k) is true.
- At the last step, you will prove that if P(k) is true then, P(k+1) is true.
How are both principles Different?
- The first principle is often considered as a weak one while the second principle is a strong principle of mathematical induction.
- In the first principle, P(n) is assumed and in the second principle, P(n) is assumed to be true for the whole set of natural numbers which is less than n+1.
Discussion Exercises for RD Sharma Solutions for Class 11 Chapter 12
- In both the exercises of Chapter 12 of RD Sharma Class 11, you will be asked to solve mathematical statements with the proper applicability of principle and hypothesis.
- Just because the second principle is a strong one, that doesn’t mean it is a more authentic approach to get a solution.
- A mathematical statement needs to be understood in a clear way to come up with legit proof.
- Applying hypotheses may get complex at a time, but Instasolv will help you to make your knowledge more powerful.
- Both base and the inductive case are mandatory to solve Mathematical induction.
- An ample number of practical questions are available in exercises. Ultimately it will create expertise about the concept.
Benefits of RD Sharma Solutions for Class 11 Chapter 12 by InstaSolv
InstaSolv has briefly explained the history of Mathematical Induction and its basic properties through RD Sharma Class 11 Maths Chapter 12 Solutions. Mathematical induction solves many types of discrete mathematical identities. Mainly with the binomial coefficient. With the help of induction, many statements can be concluded as true just by checking base and inductive steps.
Well-organized RD Sharma Solutions will enable you to memorize the function and principle of induction in a simpler way. Proving the mathematical statement and concluding them is a final step where Instasolv will help you solve all the questions correctly. Base and inductive steps require a bit of reasoning. InstaSolv is prepared well to support you achieve that. RD Sharma Class 11 Chapter 12 is a great opportunity to score well in the examination.