Consecutive Integer Addition

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

Brain saying "WOW!"
Added IntegersSumsDifferences of the Sums(Absolute Value)
0+1 1 2
1+2 3 2
2+3 5 2
3+4 7 2
4+5 9 2
5+6 11 2
6+7 13 2
7+8 15

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!

Added IntegersSumsDifferences of the Sums(Absolute Value)
0+1+2 3 3
1+2+3 6 3
2+3+4 9 3
3+4+5 12 3
4+5+6 15 3
5+6+7 18 3
6+7+8 21 3
7+8+9 24

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:

Added IntegersSumsDifferences of the Sums(Absolute Value)
-3+-2 -5 2
-2+-1 -3 2
-1+0 -1 2
0+1 1 2
1+2 3 2
2+3 5  

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.

Added IntegersSumsDifferences of the Sums(Absolute Value)
20+21+22+23 86 4
21+22+23+24 90 4
22+23+24+25 94 4
23+24+25+26 98 4
24+25+26+27 102 4
25+26+27+28 106

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

Added IntegersSumsDifferences of the Sums(Absolute Value)
20+21+22+23+24 110 5
21+22+23+24+25 115 5
22+23+24+25+26 120 5
23+24+25+26+27 125

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!

Miss One says "You can add as many integers in a row as you like, but they must all be consecutive for this trick to work! And having only one integer in the 'Added Integers' column is out of the question!"

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© Derek Cumberbatch