Mathematical induction is one of the techniques, which can be used to prove a variety of mathematical statements that are formulated in terms of n, where n is a positive integer.
Let P(n) be given statement involving the natural number n such that
(i) The statement is true for n = 1, i.e. P(1) is true.
(ii) If the statement is true for n = k (where k is a particular but arbitrary natural number), then the statement is also true for n = k + 1 i.e. truth of P(k) implies that the truth of P(k + 1). Then, P(n) is true for all natural numbers n.
Mathematical Induction is a specific technique that is primarily used to prove a given statement or a theorem. The crucial part of this method is that the theorem should stand true for every natural number. It is a useful method as you do not have to go round solving an equation or a statement for every possible value it can take.
With Mathematical Inductions, solving numerically related to series becomes convenient, and as a result, it has extensive scopes of applications in real life as well. Computer science is one of the most renowned spheres, where it is widely used.
Consequently, appropriate guidance of the method is necessary right from school-level, to enable students to make the most of these methods in simplifying the solutions.
It is the reason that our subject experts have drafted the Mathematical Induction Notes to help you understand the topic precisely. Revising from our notes before your exam will assist you in recalling the crucial topics that you might have forgotten or missed while studying the chapter.
Students need to be familiar with the following pointers when studying Mathematical Induction:
Proving a given statement or illustration is the primary motive.
The proof should stand true for all values consisting of natural numbers.
The statement should be true for the initial value as well.
It should be true for all other values till nth iteration.
Every step involved in the proof must be justified and true.
You should have the necessary reasoning and logical skills to ace the numerical related to this chapter. They should be aware of the properties of natural numbers too before delving into solving numerical.
Principle of Mathematical Induction – Revision Notes
Here is a brief description of the topics covered in this chapter which should be considered by students, when they revise our Mathematical Induction Notes.
Mathematical Induction Class 11 Notes - Principle of Mathematical Induction
There are two principles of Mathematical Induction, which you need to know right from the beginning. They are:
Deduction
Induction
This chapter deals with the latter principle. As you go through our revision notes, you will know that the former principle is based on a generalization of certain specific instances, while the latter is opposite of it. Induction is more like particular instances of generalization.
You will find a simple example in our notes of Principle of Mathematical Induction Class 11 to facilitate a more fundamental understanding of both the concepts. As a result, you will also be able to differentiate between the two. The given example is as follows:
Example:
Deduction: In here, you are provided with some facts or statements, out of which you are required to deduce or derive a particular statement or information. It is similar to deciphering the required information from the data.
For instance, consider this example-
Statement 1: Rohit is a man.
Statement 2: All men eat food.
Conclusion: Rohit eats food.
As you can see in our Mathematical Induction Notes, the two statements above help in deriving the conclusion that says Rohit is one of the men who eats food.
Induction: Herein, you will be provided with statements that signify specific instances, out of which you are likely to draw conclusions related to generalisations.
Consider this example-
Statement 1: Rohit eats food.
Statement 2: Vikas eats food.
Statement 3: Rohit and Vikas are men.
Conclusion: All men eat food.
As can be seen, you can derive the conclusion out of the three given statements which are specific instances. Therefore, you can induce a generalised statement from here. Make sure you go through each line in our Mathematical Induction Notes to ensure you gain clarity on both the topics.
In addition, we have also presented a mathematical deduction for the same. It will help you understand the concepts in a more natural way. Since the main objective is to solve numerical with the help of this method, we have presented the same with the support of algebraic representations.
For instance,
If a statement is valid for a value of n, where n = 1,
And again, the same statement is true another value, say n = k,
Once again, it is valid for n = k +1,
Therefore, one statement is true for all the above-mentioned values; as a result, it is plausible that it will be valid for all other values of n, provided they are natural numbers.
Refer to our Mathematical Induction Notes for a more logical explanation of the same. We have made a diagram with proper labelling that sums up the original idea behind induction and deduction.
Steps in Mathematical Induction
There are five steps in all you need to follow while solving a numerical using mathematical induction. In our revision notes, the steps are explained in brief so that you can easily recall them and solve the numerical.
Students should be aware that these shortcut techniques are mainly for fulfilling the purpose of simplifying the process of memorising the method of Mathematical Induction. Hence, they need to have clarity on the topics in details at the first place after which they can refer to our revision notes before the exam.
Although the diagrammatic representation in the previous section can help them in understanding the concerned topic quickly, they should not miss out any pointer written in the notes. Each tip given in our Class 11 Maths Chapter 4 Notes is equally important and must be kept in mind to understand the topic in depth.
The steps involved in solving numerically related to this topic can be summarised as follows:
Step 1: Consider P(n) to be a given statement or resultant in terms of ‘n’.
Step 2: Prove P(1) to be true.
Step 3: Consider that P(k) is also true.
Step 4: Using the previous step, make sure P(k+1) is also true.
Step 5: Now, both P(1) and P(k+1) is valid.
Consequently, by the Principle of Mathematical Induction, P(n) is true for all values of natural numbers ‘n’. To ensure that students do not miss out on any significant step, we have presented the steps one after one briefly without compromising on the main content.
Simultaneously, it will also help you to boost your exam scores. Revise these Mathematical Induction Notes regularly before your actual exam, and you can add an extra edge over to the competition.
Mathematical Induction - Illustrated Example
To further simplify your understanding, we have presented an example too. Here the questions ask you to prove than 2n is greater than n for all positive integers that n can hold. As already mentioned, we follow the same steps one by one in the following manner:
Step 1: Let P(n): 2n > n
Step 2: When n =1, 21 = 2, which is greater than 1. Thus, P(1) is true.
Step 3: Assuming P(k) is valid for any natural number k, 2k > k.
Step 4: Now, you have to prove that P(k+1) is also true, as P(k) is true. Take note of the following steps given in our Mathematical Induction Notes.
We have the equation 2k > k, so we multiply each side by 2,
Then we get 21. 2k > 2. k
or, 2(1+k) > 2k
or, 2(1+k) > k + k
or, 2(1+k) > k + 1 [since, k > 1]
Therefore, it can be seen that P(k+1) is true when P(k) is true. Again, by the Principle of Mathematical Induction, P(n) stands true for all values of n which re natural numbers. These steps are illustrated in our Principle of Mathematical Induction Class 11 Notes so that you can understand the topic clearly.
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