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There are two important questions to ask to understand the Algebra of Consecutive sums. First, what is important about the first Number of the consecutive list? Second, how is the number of numbers consecutively added related to the total?
Using: 4+5+6+7 & 4+5+6 as two consecutive sums startign with 4 address the first question?
2+3+4 and 10+11+12 as 2 consecutive sums of length three(3) numbers address the second question?
I have no clue what this means
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1 = 1 (put n = 1)
1 + 2 = 3 (put n = 2)
1 + 2 + 3 = 6 (put n = 3)
1 + 2 + 3 + 4 = 10 (put n = 4)
If the first number is not 1, but 4, then the sum will be 3 * n greater
4 = 4 = 1 + 3
4 + 5 = 9 = 2 + 2 * 3
4 + 5 + 6 = 15 = 6 + 3 * 3
4 + 5 + 6 + 7 = 22 = 10 + 4 * 3
That is, if you add n consecutive numbers starting from x, then the total is (x – 1) * n + n * (n + 1) / 2. This is the answer to the first question.
e.g. think about 5 consecutive numbers starting from 6
6 + 7 + 8 + 9 + 10 = 40
(x – 1) * n = (6 – 1) * 5 = 25
n * (n + 1) / 2 = 5 * (5 + 1) / 2 = 15
25 + 15 = 40
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