# A reboot for Proofs from the Book

I have decided to include undergraduate proofs in this blog. We will be studying formal proofs starting with logic (truth tables, logical connectives, etc.) and then study different methods of proofs (direct, indirect, etc).

While studying these concepts, we will have numerous examples from different branches of mathematics particularly number theory, probability, and combinatorics.

I’m very excited to start this series, so just keep posted.

# The Sum of Two Consecutive Odd Numbers

When a is a factor of a number b, then b is divisible by a. For example, 3 is a factor of 12, so 12 is divisible by 3. We will use this concept in the proof in this post.

Notice that the sum of the following consecutive odd numbers.

11 + 13 = 24

9 + 11 = 20

25 + 27 = 52

101 + 103 = 204

Notice that the sum is divisible by 4. Now, we can make a conjecture that the sum of two consecutive numbers is divisible by 4.

Theorem: The sum of two consecutive odd numbers is divisible by 4.  Continue reading