Mathematical Induction

In mathematics, mathematical induction is a crucial concept. The mathematical induction method is useful to prove any given axiom about any well-organised set. Usually, one can use it to verify results or establish statements formulated in terms of n, where n is a natural number. 

In the following sections, you will find brief information on the concept of mathematical induction, proof of mathematical induction, and so on.

Concept of Mathematical Induction

As mentioned in the introduction, the mathematical induction method is used for proving any given axiom about any well-organised set. Generally, it is used for verifying results or establishing statements formed in the ‘n’ terms. Note that here, ‘n’ is a natural number. The method involves a few steps given below to prove a statement, P(n):

• Verify that the statement is right in trivial cases, such as n = a, i.e. P(a) is correct. (base case)
• Assume the statement is right for n = k for some k > a, so then P(k) is correct. (inductive hypothesis)

The truth of P(k) implies the truth of P(k + 1), so the statement P(n) is true when n ≥ a.

Together, the base step and inductive step proves that P(k) => P(k + 1) => P(k + 2) Therefore, P(n) is correct for each n ≥ a. 

Here is an example of mathematical induction: 

A domino is knocked down in succession after it falls. Let’s say the second domino is kicked by the first, the second knocks down the third, and it keeps on continuing. All the dominoes will be knocked at a certain point.

Certain conditions have to be met that are the same as the steps of mathematical induction discussed earlier:

• The first domino must fall to begin the knocking process. This is the first step.

• Any two adjacent dominoes must have the same distance between them. If not, a domino may fall without knocking over the next one. This would stop the sequence of reactions. 

Here is the proof of mathematical induction: 

Given the statement P(n) involves the natural number “n” such that

• It is true when n = 1, i.e., P(1) is true 
• If it is true when n = k (here, k is a positive integer). In such conditions, it must also be true for n = k + 1, i.e., the truth of P(k) implies the truth of P(k + 1). 

Interesting Facts about Mathematical Induction

Proof by mathematical induction was not invented by a particular person at a specific time, unlike most concepts and methods in mathematics or any other subject. Whereas the principle of mathematical induction was ‘The Pythagoreans’. According to legend, Blaise Pascal is credited with originating the principle of mathematical induction. 

For all natural numbers n, P(n) is true. 

It simply states the fact. In some cases, a statement may be true for n ≥ 4. In this case, step 1 will start from n = 4, and we will verify the result for n = 4, which is P(4). This is referred to as the conditional property. 

To prove that the property holds, only demonstrate the conditional proposition: if the statement is true for n = k, then it is also true for n = k + 1. This step is sometimes referred to as induction. 

In this inductive step, the inductive hypothesis states that the given statement is true for n = k.

Sometimes in mathematics, the formula can appear in the following pattern:

1 = 12 = 1

4 = 22 = 1+3

9 = 33 = 1+3+5

This pattern indicates that 1 + 3 + 5 + 7 + … (2n – 1) =n2, i.e., the sum of the first n odd numbers is the square of n. Let us write P(n) as follows: 1 + 3 + 5 + 7 + … + (2n – 1) = n2. 

A proof that uses mathematical induction must be true for all n. The first step in proving that P (1) is true is to prove that P (2) is true. This is the basic step. P(1), therefore, states that 1 = 1. 

Inductive reasoning is the next step. We assume P (k) is true to a positive integer k. We need to prove that P (k + 1) is true. Since P (k) is true, we have 1 + 3 + 5 + 7 + … + (2k – 1) = k2 … (1) 

Consider 1 + 3 + 5 + 7 + … + (2k – 1) + {2(k +1) – 1} … (2) 

= k2 + (2k + 1) = (k+1)2 [Using (1)] 

Therefore, P (k + 1) is true, and the inductive proof is complete. 

Conclusion 

When you study mathematics, mathematical induction is one of the most basic terms taught in earlier classes. It is a method used for proving any given axiom any well-organised set. Mainly, it is used for establishing statements or verifying results formed in the ‘n’ terms. 

In this article about mathematical induction, we have studied the concept in length. We have also covered several other topics, such as a mathematical induction formula, facts about mathematical induction, and other related topics. 

P(n) is therefore true for all the natural numbers n.