9.4 mathematical formula

In this section we will study a mathematical proof called mathematical induction. 

The principles of mathematical induction: 

Let Pn be a statement involving the positive interger n. If

1. P1 is true 

2. Pk is assumed to be true

3. P(k+1) is proved to be true 

Then Pn must be true for all positive integers n

 Ex 1:

Find P(k+1) for Pk:Pk= [k^2 (k+1)^2]/4

P(k+1):P(k+1)= (k+1)^2(k+2)^2/4

 Ex 2: 1+3+5+...+(2n-1) = n^2

 Step 1 prove the statement is true for n= 1

            S1= 1

 Step 2 assume true for n= k

           1+3+5+...+(2k-1) = k^2

Step 3 prove true for n= k+1

           1+3+5+...+(2(k+1)-1) = (k+1)^2

           1+3+5+...+(2k-1) +(2(k+1)-1) =( k+1)^2

           Substitute the assumption for n= k

           Get k^2 +(2(k+1)-1) = ( k+1)^2

                  K^2+ 2k + 1= K^2+ 2k + 1

          Therefore the statement is true for all n as in positive integers.