Exercise 29 Page 457

Let's attempt to find a pattern by calculating the sum of two odd integers. 1+3&=4 &&3+5=8 5+7&=12 &&9+11=18 Examining the sums, we see that all of them are even. We can make a conjecture by inductive reasoning. Conjecture: & The sum of two odd &integers is always even. To use deductive reasoning, we have to prove this is true for all integers. If we let a and b be arbitrary integers, then 2a and 2b will always be even, since all even integers are divisible by 2. To make these into odd integers, we add 1 to them. With this information we can write two arbitrary odd integers. Odd Number1: 2a+1 Odd Number2: 2b+1 Let's add them and see what happens.

As we can see, the sum of two arbitrary odd numbers can be rewritten as a product where one factor is 2. As already explained, the definition of an even number is that it is divisible by 2, which means our conjecture is true.