The principle of mathematical induciton is one of the important topics in pure and applied mathematics. The principle of mathematical induction is an interesting method of proof in Mathematics. The principle of mathematical induction is used to prove a mathematical statement for all possible cases. It has applications in pure and all branches of mathematics.
JEE Main: Study Materials | High Scoring Topics | Preparation Guide
JEE Main: Syllabus | Sample Papers | Mock Tests | PYQs
This article is about the concept of Principle of Mathematical Induction class 11 Principle of Mahthematical Induction chapter is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE, BCECE, and more.
Mathematical induction is a method or technique used to prove a mathematical statement generalized for
From this, we could say that sum of 2 odd numbers is
Now, let us state the principle of mathematical induction.
The following result is also called the first principle of mathematical induction.
Let
The statement is true for
If the statement is true for
Then,
1. This is the first step, typically a fact about the statement called the base step. Here, the mathematical statement is checked whether it is true for
2. In the second step, it is assumed that the statement is true for
Now, let's apply this principle of mathematical induction to our well known result, the sum of
Let
Substituting the value of
Let us assume that the statement is true for
We need to show that
That is,
This implies,
The principle of mathematical induction proof is direct and obvious. To check whether a statement is true for
After which every
Let
The statement is true for
If the statement is true for
Then,
The difference between the first principle of mathematical induction and second principle of mathematical induction is that, in first principle of mathematical induction, the inductive step is assumed for
Now, let's look into second principle of mathematical induction examples.
Every integer
For
Now, Assume that every integer
Now we can prove the statement for
Case 1: If
Case 2: If
Thus, by second principle of mathematical induction, every integer
Example 1: The smallest positive integer
1)
2)
3)
4)
Solution:
Let
Step I: For
Step II : Assume that
Step III : For
and
Which is true, hence (ii) is true.
From (i) and (ii),
Hence this is true. Hence by the principle of mathematical induction
By checking options,
(a) For
(b) For
(c) For
(d) For
But the smallest positive integer n is 2. Hence, the answer is the option 2.
Exapmle 2: Let
1)
2)
3)
4) Principle of mathematical induction can be used to prove this formula
Solution:
Now,
Which is NOT true
So, option C, D are wrong
This is also wrong
Now,
Assuming
Then
So,
Example 3: By the principle of mathematical induction, prove that, for all integers
Solution:
Let,
Substituting
Let us assume that the statement is true for
We need to show that
That is,
This implies,
Example 4: Let
1)
2)
3)
4)
Solution:
We will use the principle of mathematical induction to solve this problem.
The base case is when
Therefore,
is true for
We need to show that
Expanding the left-hand side using the binomial theorem,
Since
Therefore, we have:
Therefore, the statement is true for all positive integers
Hence,
is true for all positive integers
Example 5: Which of the following statements is true for all positive integers
1)
2)
3)
4)
Solution:
We will use the principle of mathematical induction to solve this problem.
For
which is true.
Therefore,
Assume that the statement is true for some arbitrary positive integer k .
We need to prove that the statement is also true for
Expanding
Now, using the inductive hypothesis
To prove that
We can see that this inequality holds for all positive integers
Therefore, the inductive step is proved.
Hence, by the principle of mathematical induction, we can say that
for all positive integers n .
Algebra at the JEE level is very interesting. All topics are more or less independent of each other. And one of the interesting and important topics is Principle of mathematical induction class 11 and every year you will get 1-2 question in JEE Main exam as well as in other engineering entrance exams. JEE question paper is highly unpredictable, you never know questions from which topic will be asked. A general trend noticed in Mathematics paper is that a question involving multiple concepts are asked. As compared to other chapters in maths, Principle of Mathematical Induction questions for JEE MAINS requires less effort to prepare for the examination.
Principle of Mathematical Induction is one of the easiest topics, you can prepare this topic without applying many efforts. Start with understanding the method of principle of mathematical induction. Practice many problems from each topic for better understanding. Practice from the previous year Principle of Mathematical Induction questions for JEE MAINS.
If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.
NCERT Notes Subject Wise Link:
Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.
Then you can refer to the book Arihant Algebra Textbook by SK Goyal or Cengage Algebra Textbook by G. Tewani but make sure you follow any one of these not all. Principle of Mathematical Induction is explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.
NCERT Solutions Subject wise link:
Let
The statement is true for
If the statement is true for
Then,
Principle of mathematical induction is a method or technique used to prove a mathematical statement. There is no specific formula for principle of mathematical induction. The statement of principle of mathematical induction is "Let
The statement is true for
If the statement is true for
Then,
The difference between the first principle of mathematical induction and second principle of mathematical induction is that, in first principle of mathematical induction, the inductive step is assumed for
Let
The statement is true for
If the statement is true for
Then,
The principle of mathematical induction is used to prove the mathematical statements for
12 Feb'25 01:33 AM
20 Dec'24 04:39 AM