The Principle of Mathematical Induction

Definition of Mathematical Induction:

Let us learn about the meaning of mathematical Induction in detail.
Simplest and general form of mathematical induction proofs that statement relating natural number ’n’ holds for every values of ‘n’.

Induction proof has two steps:

1. Basis

2. Inductive step

While understanding the meaning of mathematical induction we must keep in mind about the Principle of Mathematical Induction.

The Principle of Mathematical Induction:

Related to all positive integer ‘n’ let be statement or proposition P(n).

If P(1) true, and P(k + 1) is true whenever P(k) true, then P(n) is true for all positive integers ‘n’.
Working Rules for Using the Principle of Mathematical Induction:

Step (1): Show that the result is true for n = 1.

Step (2): Assume the validity of the result for ‘n’ equal to some arbitrary but fixed natural number, say ‘k’.

Step (3): Show that the result is also true for n = k + 1.

Step (4): Conclude that the result holds for all natural numbers.
Example Sum for Principle of Mathematical Induction

Ex 1: Prove by the principle of mathematical induction method 4 + 8 + 12 + … + 4n = 2n(n + 1)

Solution: P(n)= 4 + 8 + 12 + … + 4n = 2n(n + 1)

Put n = 1

P(1) true

Assume that statement be true for n = k

Assume P(k) true.

Assume 4 + 8 + 12 + … + 4k = 2k(k + 1) be true

To prove P(k + 1) - true.

4 + 8 + 12 + … + 4(k+1) = 2(k+1)(k+1 + 1)= 2(k+1)(k+2)

P(k + 1) is true.

If P(k) true, then P(k + 1) is also true.

P(n) is true for all n [in] N.

Ex 2: Prove by the principle of mathematical induction method n2+ n is even.

Solution: P(n)=n2+ n is even

n2+ n = 1

n = 1

2+ 1= 2

P(1) true

Assume that statement be true for n = k

Assume P(k) true.

To prove P(k + 1) true.

To prove (k + 1)2 + (k + 1) even

(k + 1)2 + (k + 1) = k2+ 2k + 1 + k + 1

= k2+ 2k + k + 2

= an even number

P(k + 1) true.

Thus if P(k) true, then P(k + 1) also true.

P(n) true for all n [in] N.

n2+ n even for all n [in] N by the principle of mathematical induction.


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