chapter mathematical induction

1+3+5+7+......+(2n-1)=

Dear student 

Proof by induction on n:
 
Step 1:  prove that the equation is valid when n = 1
 
              When n = 1, we have (2(1) - 1) = 12, so  the statement                        holds for n = 1.
 
Step 2:  Assume that the equation is true for n, and prove that the                   equation is true for n + 1.
 
              Assume:  1 + 3 + 5 + ... + (2n - 1) = n2
 
             Prove:  1 + 3 + 5 +...+ (2(n + 1) - 1) = (n + 1)2
 
                 Proof:  1 + 3 + 5 +... + (2(n + 1) - 1) 
                              = 1 + 3 + 5 + ... + (2n - 1) + (2n + 2 - 1)
                              = n2 + (2n + 2 - 1)  (by assumption)
                              = n2 + 2n + 1
                              = (n + 1)2
 
So, by induction, for every positive integer n,
1 + 3 + 5 + ... + (2n - 1) = n2.