Class 11 Maths Mathematical Induction Principle of Mathematical Induction

The Principle of Mathematical Induction

Deduction: Generalization of Specific Instance

• Example : Rohit is a man & All men eat food à Rohit eats food.
• Example : Mukesh is an Engg & All Engg earn good money à Mukesh earn good money.
• Example : Sun is a star & All stars have their own light à Sun has its own light.

Induction: Specific Instances à Generalization

• Rohit eats food. Vikash eats food. Rohit and Vikash are men à All men eat food
• Statement is true for n=1, n=k & n=k+1 à Statement is true for all.

For a statement P(n) involving the natural number n , if

1.  P(1) is true
2.  Truth of P(k) implies the  truth of P (k + 1).

Then, P(n) is true for all natural numbers n.

Property (i) is simply a statement of fact.

Property (ii) is a conditional property. It does not assert that the given statement is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1.  This is sometimes referred to as inductive step. The assumption that the given statement is true for n = k in this inductive step is called the inductive hypothesis.

.