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.

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