Denote the sum by <em>S</em>. So
<em>S</em> = 5 + 11 + 17 + 23 + ... + 83
There's a constant difference of 6 between consecutive terms in <em>S</em>, so the 3 terms before 83 are 77, 71, and 65. So
<em>S</em> = 5 + 11 + 17 + 23 + ... + 65 + 71 + 77 + 83
Gauss's approach involves inverting the sum:
<em>S</em> = 83 + 77 + 71 + 65 + ... + 23 + 17 + 11 + 5
If we add terms in the same position in the sums, we get
2<em>S</em> = (5 + 83) + (11 + 77) + ... + (77 + 11) + (83 + 5)
and we notice that each grouped term on the right gives a total of 88. So the right side consists of several copies <em>n</em> of 88, which means
2<em>S</em> = 88<em>n</em>
and dividing both sides by 2 gives
<em>S</em> = 44<em>n</em>
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Now it's a matter of determining how many copies get added. The terms in the sum form an arithmetic progression that follows the pattern
11 = 5 + 6
17 = 5 + 2*6
23 = 5 + 3*6
and so on, up to
83 = 5 + 13*6
so <em>n</em> = 13, which means the sum is <em>S</em> = 44*13 = 572.
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