JEE Main Important Chapter - Mathematical Induction

A methodology or approach for proving a statement, theorem, or formula that is thought to be true for any natural number N is known as mathematical induction. The non-zero numbers that are utilised for counting are known as natural numbers. The concept of natural numbers is represented by the letter N. Natural numbers are not negative and do not contain fractions. The 'Principle of Mathematical Induction' is a generalisation of this that may be used to prove mathematical statements.

For example, the assertion $1^3 + 2^3 + 3^3 + ..... +n^3 = \left(\dfrac{n(n + 1)}{2}\right)^{2}$ is valid for all natural number values.

Therefore, Mathematical Induction is a method for proving natural number conclusions or establishing claims.

JEE Main Maths Chapters 2024 

Important Topics of Mathematical Induction

• Mathematical Induction

• Principle of Mathematical Induction

What is Induction?

Mathematical induction is like a way of proving things in math, especially when we're talking about numbers. It's a bit like how you'd knock over a line of falling dominoes. When you push the first domino, it falls, and if each domino is set up right, the next one falls, and then the next one, and so on. In math, it's a bit like saying if something is true for one number, it's true for the next one, and the one after that, and so on. It's a way to be sure about things when dealing with numbers.

First Principle of Mathematical Induction

Mathematical induction is a powerful technique used to prove mathematical statements, It is especially useful for proving statements that are true for all positive integers.

To use mathematical induction to prove a statement, we need to follow two steps:

• Base Case: This is the step where we prove that the statement is true for the smallest positive integer, which is 1.

• Prove the inductive step. This is the step where we assume that the statement is true for some positive integer k, and then we prove that it is also true for k + 1.

If we can prove both the base case and the inductive step, then we can conclude that the statement is true for all positive integers.

Second Principle of Mathematical Induction

Second Principle is greater than the first principle. It is also known as strong induction. In a special case, no base case is required; otherwise, multiple base cases may be required. In order to prove that k +1 is true, we must first prove that the statement is correct for all numbers less than k+1.

Important Concepts of Mathematical Induction

Principle of Mathematical Induction

Every non-negative integer belongs to the class F if integer 0 belongs to the class F and F is hereditary. If integer 1 is a member of the class F and F is hereditary, then every positive integer is a member of F.

The principle is also frequently expressed in the intensional form: A property of integers is said to be hereditary if any integer x has the property and so does its successor. If a property of the numeral 1 is hereditary, then every positive integer possesses that feature as well.

Principle of Mathematical Induction Solution and Proof - Mathematical Induction Steps

Consider P(n) is an example of a statement, where n is a natural number. Then apply the following approach to determine the validity of P(n) for each n:

Step 1: Verify that the given statement is correct when n = 1.

Step 2: Assume that the above assertion P(n) holds for n = k, with k being any positive integer.

Step 3: For any positive integer k, prove that the result is true for P(k+1).

If the preceding requirements are met, it can be argued that P(n) is true for all n natural integers.

Note: 

When we reach the inductive step in the mathematical induction principle, we must assume p (k) in order to prove the statement for p (k+1). This is known as the induction hypothesis.

Properties of Mathematical Induction

The properties that can be proven using the Mathematical Induction Principle are listed below.

1. $1+2+3+\ldots+n=\dfrac{n(n+1)}{2}, n \geq 1$

2. $1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\dfrac{n(n+1)(2 n+1)}{6}, n \geq 1$

3. $1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\dfrac{n^{2}(n+1)^{2}}{4}, n \geq 1$

4. $1+3+5+\ldots+(2 n-1)=n^{2}, n \geq 1$

5. $2^{n} > n, n \geq 1$

Proof by Mathematical Induction

The concept begins with a factual statement and ends with a conditional assertion. According to this, if the given assertion is true only for some positive integer k, then the statement P(n) is valid for n = k + 1.

This is also known as the inductive step, and the inductive hypothesis is the assumption that P(n) is true for n=k.

Example:

The proof that the sum of the first n odd positive numbers equals $n^2$ that is,

$1 + 3 + 5 + \ldots + (2n - 1) = n^2$ …….. (1)

Let F denote the class of integers for which equation (1) holds; then integer 1 belongs to F because $1^2 = 1$. Now, If integer $x$ belongs to F then,

$1 + 3 + 5 + \ldots + (2x − 1) = x^2$ …….. (2)

After $2x - 1$, the next odd integer is $2x + 1$, which when added to both sides of equation (2) gives

$1 + 3 + 5 + \ldots + (2x + 1) = x^2 + 2x + 1 = (x + 1)^2$

Equation (2) is the hypothesis of induction and it states that equation (1) holds when n equals x, whereas equation (3) states that equation (1) holds when n equals x + 1. Since equation (3) follows from equation (2), it has been established that whenever x belongs to F, the successor of x also belongs to F. As a result of the mathematical induction principle, all positive integers belong to F.

Application of Mathematical Induction in Real Life

Mathematical induction can be compared to dominoes falling. When a domino falls, it causes the following domino to fall in turn. The first domino takes down the second, and the second falls down on the third, and so on. All of the dominoes will be knocked over in the end. However, there are several rules that must be followed:

1. To start the knocking process, the first domino must fall. This is the first and most important step.

2. Any two adjacent dominoes must have the same space between them. Otherwise, a domino could fall without knocking down the next. The chain of reactions will then come to an end.

3. Maintaining the same inter-domino distance ensures that $P(k) \Rightarrow P(k + 1) $ for each integer $k \geq a$. This is when the inductive process begins.

Example: Dominoes Falling

How is the First and the Second Principle of Mathematical Induction Similar?

The First and Second Principles of Mathematical Induction are quite similar and yield the same results. Since both methods produce the same outcomes, you can transform a proof done using one method into the other and vice versa.

For instance, suppose you have a proof for a statement P(n) using the Second Principle (strong induction). Now, let's define R(n) as "P(m) is true for all m, where 0 ≤ m ≤ n." If R(n) is true for all n, it implies that P(n) is also true for all n. You can easily adapt your proof from P(n) to R(n) using the First Principle (weak induction).

Conversely, if you had initially proven P(n) using the First Principle (weak induction), it's already a proof using the Second Principle (strong induction). In the base case, P(0) is proven without making any assumptions, and in the inductive step, when proving P(n + 1), you can assume all earlier cases (P(0), P(1), ... P(n)), but you only need to use the P(n) case.

In simple terms, both principles of induction are equally effective, and you can switch between them when proving mathematical statements.

Difference between the First and the Second Principle of Mathematical Induction

JEE Main Maths - Mathematical Induction Study Materials

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Conclusion 

The chapter on Mathematical Induction is a vital stepping stone for JEE Main aspirants. It equips students with a powerful tool to prove statements, theorems, and inequalities for a wide range of mathematical problems. This knowledge extends beyond exams, as it lays the foundation for advanced mathematical concepts and problem-solving techniques. By grasping the principles of induction, students are not only prepared for their JEE Main examinations but also for a future filled with mathematical challenges. It instills logical reasoning, precision, and the ability to construct rigorous proofs, enabling students to excel in various scientific and engineering disciplines.