[Maths Class Notes] on Mathematical Induction Pdf for Exam

Understanding the Principle of Mathematical Induction

Proof by Induction will help you understand the meaning of mathematical induction. It consists of –

1) The basis or base case proves that statement for n = 0 without assuming knowledge of other cases. 

2) The 2nd case or the inductive step proves if the statement holds for any given case n = k, it must also hold for the next case n = k + 1.

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Application of Mathematical Induction in Real Life – ‘The Domino Effect.’ 

If you queue a thousand dominoes and want to let them all fall by allowing the first domino to fall, how would you queue it?

The best idea is to queue it such that:

1. When the very first domino topples, it will lean against the second domino and make it fall.

2. Ensure that each domino will hit the domino next to it and that each hit makes a domino fall.

3. If conditions (1) and (2) are satisfied, then all the dominoes will fall, proving the principle of mathematical Induction.

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Mathematical Induction Example 

Here is a mathematical induction question:

Problem 1: Family Tree 

Imagine that your great, great, … grandfather Jim had two children. Call this generation one, so that generation one contains 2 offsprings of Jim Each of these children has two children, so generation two will have 4 offsprings. Each of the four offspring has 2 children, so, at 3rd generation, we have eight offspring. If this sequence continues generation after generation, prove that at n generation, there will be 2n offsprings?

Solution:Base Case: P(1) asserts that generation 1 has 21 offspring, which is true since we are told that Jim had two children.

Inductive Step: We assume for an arbitrary integer, say k, P(k) is true. That means, that generation k has 2k offspring. We have to show that k + 1 generation has 2k + 1 offspring.

Generation k + 1 has 2 x 2k offsprings [Generation k has 2k assumed to be true]

Generation k + 1 has 21 x 2k offsprings

Generation k + 1 has 2k + 1 offsprings

Thus, P(k + 1) is true because P(k) is also true. 

Hence P(n) is true for all natural numbers.