MATHEMATICAL INDUCTION

 PRINCIPLE OF MATHEMATICAL INDUCTION 

The process of drawing a valid general result from particulat result is called the process of induction.

The principle of mathematical induction{PMI} is a mathematical process which is used to establish the validity of a general result involving natural numbers. 

Statement 

A sentence is called statement if it is either true or false but not both.

Example-"two plus five is equal to seven"" is a statement because the sentence true.

A statement concerning the natural  number 'n'  is generally denoted by (n)

Example- If P(n)) denotes the statement : ''n(n+1) is an even number'', then 

P(2)=2(2+1)=6 is an even number and                                                                                      P(5)=5(5+1)=30 is an even numer etc..                                                                                      here P(2) & P(5) both true.

THE PRINCIPLE OF MATHEMATICAL INDUCTION 

If P(n) is a statement (n ∈ N )  such that :

1) P(1) is true and  

2) true of P(k) implies the truth of P(k+1): then by principle of mathematical induction the statement P(n) is true for all n ∈ N.

WORKING RULES 

Step 1 Show that the given statement is true for n=1
Step 2 Assume that the statement  is true for n=k
Step 3 Using the assumption made in step 2, show that the statement is true for n=k+1
Step 4 This completes the principle of mathematical induction. Conclude that the given statement is true for all n ∈ N.

EXAMPLE – PROVE THAT 2+4+6+………+2n=n(n+1), n ∈ N

SOLUTION – THE GIVEN STATEMENT 2+4+6+………+2n=n(n+1), n ∈ N

We shall show that (1) is true for n ∈ N , by using PMI

STEP 1 Let n=1,  then L.H.S of (1) =2 R.H.S of (1)= 1(1+1)=2

(1)  is true for n=1

STEP 2 Let the statement be true for n=k

2+4+6+………+2k=k(k+1)

STEP 3 Let n=k+1, then L.H.S of (1)=2+4+6+…2(k+1)

    =2+4+6+………+2k+2(k+1)

=(k+1)(k+2)

Also R.H.S of (1) =(k+1)(k+1+1)=(k+1)(k+2)

Hence (1) is true for n=k+1. PMI, 2+4+6+………+2n=n(n+1), n ∈ N