Question

lWhat is the principle of mathematical induction...... explain with an example....

Answer 1

$\huge{\mathbb{\red{ANSWER}}}$

Let P(n) be a mathematical statement about non-negative integers n and n be a fixed non-negative integer.

(1) Suppose P(n₀)is true i.e.. P(n) is true for n = n₀.

(2) Whenever k is an integer such that k ≥ n₀ and P(k) is true, then 
P(k + 1) is true.


....
example...

(attachment)..


$\mathbb{\blue{@\:EMINATOR}}$

Answer 2

The Principles of mathematical induction are :

P ( 1 ) is true .

P ( 2 ) is true .

P ( 3 ) is true .

................

Let P ( n ) = true for a natural number n .

Then P ( n + 1 ) = true whenever P ( n ) = true .

For example :

Example 1 :

Prove by induction that 1 + 3 + 5 .... + ( 2 n - 1 ) = n² for all n ∈ N

Approach :

P ( 1 )

= > 1 = 1² = true

P ( 2 )

1 + 3 = 2²

= > 4 = 2² = true .

Let P ( m ) = true

= > P ( m ) = 1 + 3 + 5 ..... ( 2m - 1 ) = m²

Then For P ( m + 1 )

= > P ( m + 1 ) = 1 + 3 +............. ( 2 ( 2 m + 1 ) - 1 )

= > P ( m + 1 ) = 1 + 3 ........... + ( 4 m + 2 - 1 )

= > P ( m + 1 ) = 1 + 3 ............. + ( 2 m - 1 ) + 2 m + 1

= > P ( m + 1 ) = m² + 2 m + 1

= > P ( m + 1 ) = ( m + 1 )²

Hence it is proved !