Question

prove that an odd positive number is of the form 4k+1 or 4k+3

Answer 1

p(n) = 4k + 1                                                              

For n=1 is true, since                                                                                                                                                  n=1;                                                                           
p(1) =4(1)+ 1;
          5//
suppose  is true for some  n= k >+1,that is
p(k) =4k + 1 = o ( oddnumber)--------(1)
 Prove that  is true for n = k + 1, that is
p(k+1) = 4(k+1) + 1
           =4k +4 +1
           =4k +1 + 4
             =o-1+1+4
             =o+4               (If n is odd and m is even, then n + m is odd )
accordingly         
 thereby showing that indeed P(k + 1) holds.

Since both the basis and the inductive step have been performed, by mathematical induction, the statement P(n) holds for all natural numbers n