Consecutive Integer Addition

If you add sets of consecutive integers in a list, something special can happen!

Note: Remember that absolute value is always positive, so that subtracting bigger numbers from smaller numbers won't give you a negative number. You should already know that the answer to a subtraction problem is called a difference.

Let's add 3 integers in each row!

The differences of the sums tell you how many integers you added up! Furthermore, it's the same number in each row!

Here's what happens if you include negative integers:

Have you noticed the sums are consecutive odd numbers? This only happens when you add 2 consecutive integers. But if you add 3 consecutive integers, then you get multiples of 3. Let's see what happens if you add 4 consecutives starting at the number 20.

Now the sums skip 4 integers each; that's the number in every cell of the difference column. What if we add 5 integers?

Now the sums are multiples of 5; again, that's the number in every cell of the difference column. Sometimes, you get multiples of the difference in the sums column; other times, the sums skip that many integers, if you add 4 consecutives or more. To get true multiples of the difference, you must add an odd number of integers; if you add an even number of integers, then you won't get multiples of the difference, but the sums will skip that many integers. All of this is practically a bunch of summations!

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© Derek Cumberbatch