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

Square of Odd Numbers Minus 1

Many numbers have interesting properties worth investigating. These properties can be used to introduce the notion of proofs. Teachers may try this out in middle school or high school students. Although the proofs might not be as elegant or as formal such as written in books, its notion and the reasoning behind it is more important. Let us try to answer the question below.

Investigate the numbers which are one less than the square of odd numbers.

We can try some examples and see their properties.

$latex 3^2 – 1 = 8$

$latex 5^2 – 1 = 24$

$latex 7^2 – 1 = 48$

$latex 9^2 – 1 = 80$

Looking at the pattern, it is quite obvious that all of them are divisible by $latex 8$. Maybe, you might want to have more examples. To lessen the burden the calculation, you might want to use a spreadsheet to do this.  Continue reading