By using mathematical induction, prove that the product of two consecutive odd number is odd number..

We know that given any whole number n, 2n + 1 represents an odd number.

We also know that multiplying any number by 2 results in an even number                                                                             and an even number added to an odd number results in an odd number.

Proof:

Is it true when n = 1?

   1 x 3 = 3

   Yes, it is true for n = 1

Assume true for n = k; that is:  (k)(k+2) is an odd number.

Under this assumption, will it be true for k+1?

   [k+1][(k+1) + 2]  =  [k + 1][k + 3]  =  k² + 4k + 4  

                               =  k² + 2k + 2k + 4  =  (k)(k + 2) + 2(k + 2)

   Since, by assumption, (k)(k+2) is an odd number,                                                                                              and 2(k + 2) is an even number (because it is a number multiplied by 2),                                          then (k)(k + 2) + 2(k + 2) is an odd number (because it is an odd number added to an even number).

   Yes, it is true under the assumption.

Q.E.D.