Two students were discussing whether or not the sum of any three consecutive whole numbers was divisible by 3. One student said this was true because it worked for every example she had tried. The second student said it might not always be true because you can't prove something with examples. Write an explanation that begins with examples and then leads the students to understand that the sum of any three consecutive whole numbers is always divisible by 3.
Let's start by examining some examples to see if the sum of any three consecutive whole numbers is divisible by 3.
Let's take the first three consecutive whole numbers: 1, 2, and 3. If we add them up, we get 6. Now, let's try the next three consecutive whole numbers: 2, 3, and 4. Adding them up gives us 9. And let's try one more set: 3, 4, and 5. Adding them up gives us 12.
From these examples, we can see that the sum of any three consecutive whole numbers is indeed divisible by 3, as 6, 9, and 12 are all divisible by 3.
The first student believes that this pattern will hold true for all cases because it has worked for every example she has tried. However, the second student raises a valid point by stating that examples alone are not enough to prove something. So, let's explore the reasoning behind this pattern further.
To ensure that we can prove the sum of any three consecutive whole numbers will always be divisible by 3, we should employ mathematical reasoning.
We can represent the three consecutive whole numbers mathematically by using variables. Let's call the first number "n," the second number "n + 1," and the third number "n + 2."
Now, let's add up these three numbers:
n + (n + 1) + (n + 2)
Combining like terms, we have:
3n + 3
The expression 3n + 3 can be factored further:
3(n + 1)
We can see that this expression is a multiple of 3, as it is equal to 3 times the quantity (n + 1).
Since 3(n + 1) is divisible by 3, we can conclude that the sum of any three consecutive whole numbers is always divisible by 3.
Therefore, although the first student's examples were accurate, the second student was also correct in pointing out that we need to provide a logical proof beyond mere examples. By utilizing mathematical reasoning, we have now established that the sum of any three consecutive whole numbers is indeed always divisible by 3.