Consecutive Integer Addition
If you add sets of consecutive integers in a list, something special can happen!
Added Integers  Sums  Differences 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 Integers  Sums  Differences 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 Integers  Sums  Differences 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 Integers  Sums  Differences 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 Integers  Sums  Differences 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!
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© Derek Cumberbatch