13 = 1
13 + 23 = 9
13 + 23 + 33 = 36
... + 43 = 100
... + 53 = 225
... + 63 = 441
At this point, a very reasonable guess is that each sum is a perfect square. If we each align successive term of the series with the square root of the sum of the cubes to that point, we have
1 : 1
2 : 3
3 : 6
4 : 10
5 : 15
6 : 21
This should be recognizable as the simple cumulative sum of the integers in a series, the formula for which is
Sn = n(n + 1) / 2.
Therefore, it is reasonable to guess that the direct formula for the sum of cubes from 1 to n is
[n(n + 1) / 2]2.
This is what we will prove below.
Alex, thank you for doing this proof! n previous math classes I have worked with this problem, but we never proved it or even talked about the proof. This is a wonderful proof, it allows me to make connections between things some previous professors have said about it. Great Job!
ReplyDeleteI agree with Ashley. Clear and precise proof by induction. Only thing I'd add for an exemplar would be consolidation. One framework we sometimes use is what? (recap - not so necessary here), so what? (why does it matter) or now what? (given this, what do you wonder about next).
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