LECTURE 2 PRINCIPLE OF MATHEMATICAL INDUCTION CHAPTER 6 CLASS 11

Introduction to the Principle of Mathematical Induction
The Principle of Mathematical Induction (PMI) is a powerful method of proof used to establish the truth of an infinite number of statements, typically indexed by natural numbers. It helps us prove that a statement is true for all natural numbers, starting from a base case and then showing that if it holds for one natural number, it holds for the next one.
Key Idea: - If you want to prove a statement P(n) is true for all natural numbers n =k, where k is a given starting number, often k = 1, the process involves two main steps:
Steps in Mathematical Induction
1. Base Case: - Prove that the statement is true for the first value of ‘n’, typically n = 1 or the smallest number in the range.
Show that P(1) is true, i.e., check the validity of the statement for n = 1.
2. Inductive Step: - Prove that if the statement holds for some arbitrary n = k, then it also holds for n = k + 1.
Assume that P(k) is true for some arbitrary k, this assumption is called the “inductive hypothesis”.
3. Now, prove that P(k + 1) is also true, using the assumption P(k).
Conclusion: -
If both steps are successfully carried out, the principle of mathematical induction concludes that the statement P(n) is true for all n belongs to N.
Formulas in Principle of Mathematical Induction
Some of the common types of problems that can be solved using PMI include:
(i) Sum of First n Natural Numbers: -
1 + 2 + 3 + ....... + n = n(n + 1)/2
(ii) Sum of First n Squares: -
1^2 + 2^2 + 3^2 + ...... + n^2 = n(n + 1)(2n + 1)/6
(iii) Sum of First n Cubes: -
1^3 + 2^3 + 3^3 + ...... + n^3 = [n(n + 1)/2]^2
(iv) Geometric Progression Formula: -
1 + r + r^2 + ...... + r^{n-1} = (1 - r^n)/(1 - r), where r≠1
Inequalities: - Proving inequalities such as:
2^n is greater than n^2 for all n = 5
This method is particularly helpful for proving results that are indexed by natural numbers and involve recursive patterns or series.