Principle of Mathematical Induction

Even though mathematics is mainly related to problems, calculations, and estimation, all these tasks can’t be done unless you have proof in your hand. 

For example, we all know that tan a = sin a/cos a. But can you really tell from where this formula has been derived? No right? Well, the basic trigonometric formulas are derived from right-angled triangles using Pythagoras theorem, and other modules. Similarly, if you are asked to prove that the sum of natural numbers is given by a certain expression, you will need to use the binomial or AP method. 

No matter what proof you are asked to present, one important step included is the induction step which depends on the principle of mathematical induction.

A Brief Recap of Fundamental Mathematics

Before learning about the principle of mathematical induction, it will be best if you recall some of the crucial fundamentals of Math. 

1. Natural numbers start from 1 and can continue up to n on the positive side of the number line axis. 
2. The nth term is said to be the destination with which one needs to work with a series. 
3. There are several series that can be formed using these natural numbers like square series, cube series, Fibonacci series, and so on. 
4. The sum, product, remainder, and others of natural numbers will also be a natural number. 
5. For proving any numeral equation, we can use variables where the resultant value will be stored. 

Introduction to Mathematical Induction

When you have to prove an expression, there are three main steps that need to be considered. Let’s consider an expression like X(n). Therefore, to find the value of this expression, the below steps need to be followed:

1. It needs to be verified that if we replace n in the expression with another variable like b, then X(b) will also satisfy the resultant equation defining the function X. 
2. If another variable, a is considered in a way that a >= b, then, X(a) will also become true. This is known as the inductive calculation or hypothesis because here we are assuming the variable b to satisfy the conditions of the X function. 
3. Lastly, if X(a) and X(a+1) is true, then we can say that the expression X(n) will be satisfied with any value of n, provided, n>=-b. 

If we combine the first two statements discussed above, the principle of mathematical induction can be described properly. It states that if X(a) is true, then X(a+1) and X(a+2) will also be true and the value of X(a) => X(a+1) => X(a+1).

To understand this unique evaluation technique, let’s consider the example of dominos. When the first domino is toppled, it causes a chain reaction and one by one all the dominos will fall over in a series. But this can only happen if the first domino topples over successfully and the distance between two dominos is the same throughout the entire arrangement. 

Example of mathematical induction with a few series samples

The principle of mathematical induction can be better understood with the help of an example that we have discussed in the below section. 

Example 1#

If n>=1, then we have to prove that 12 + 22 + 32 + 42 + …….. + n2 = {n(n + 1) (2n + 1)} / 6.

For doing so, let’s consider that the series 12 + 22 + 32 + 42 + …….. + n2 is expressed as X(n). Now, if n=1, the value of the X function will be stated as: 

X(1) = [1.(1+1)(2.1+1)]/6 = 1

But if we consider a variable say t whose value is not known but t is a natural number, then X(t) can be written as: 

12 + 22 + 32 + 42 + …….. + t2 = {t(t + 1) (2t + 1)} / 6

Its exact value will be given by the equation: X(t+1) = X(t) + (t+1) and the final value will be given as (t+1)((t+1)+1)(2(t+1)+1)/6.

There are two major types of mathematical induction processes based on the statements defined above: 

1. In the weak induction, X(t) will be true only when t has a definitive value. For any other value of t, this expression won’t satisfy the final theorem. 
2. In the strong induction, X(t) will be true for all possible values of t and not just a single value. 

Derivatives of the mathematical induction process

Using the concept of mathematical induction, several postulates and theorems can be proved. In this section, we will talk about two major concepts of mathematics that have gained a lot of fame in recent years- the polynomial remainder theorem and Euclid’s Postulates. 

The polynomial remainder theorem

The remainder theorem can be explained by using the concept of the Euclidean division technique. This theorem states that when a polynomial function X(n) is divided by (n-m), another polynomial is obtained with a smaller value and an additional remainder. 

This remainder obtained from the division is nothing but the value of the X(m) function. It is mainly used for the factorization of different algebraic expressions using any binomial term that cannot be done by the traditional division methods. Rather Euclid’s postulates are used to obtain the factorized expression after performing the polynomial divisions. 

For example, let’s consider an algebraic expression to be” f(x) = x3 + 14×2 + 40 that needs to be divided by (x2+4).

Or, f(x) = (x3 + 10×2 + 4×2 + 40)/ (x2+4)

Or, f(x) = [(x2+4)(x+10)]/ (x2+4)

Or, f(x) = x+10. 

Now, if we consider x2+1 = 0, then x = (-1)1/2. Putting this value in the final expression of f(x), the expression will become: 

f[(-1)1/2] = (-1)1/2 + 10

The entire above expression is defined by the principle of mathematical induction which is why this particular topic is considered to be one of the most important matters. It not only helps in defining different types of series and proving their resultant values but also provides more knowledge about the remainder theorem and Euclid’s postulates. 

Conclusion

After learning the principle of mathematical induction, you will be able to solve a wide range of problems so that you can easily solve them without much hassle. However, you must know about the different principles that have been introduced towards the advanced induction processes. Once you have the concept within your grasp, you won’t have to worry much. However, for some students, the induction method can be proved to be quite difficult because the inductive step involved in proving an expression and finding the function’s value for a given value is not easy to derive. That’s why understanding all the aspects of the principle of mathematical induction need to be understood appropriately.