1

I'm trying to wrap my head around the sum of the first $k^{th}$ powers.

I understand the sum of the first n powers of 1, $n(n+1)/2$.
In order to reach the sum of the first n squares, I attempted to square the whole thing. But I seem to have arrived at the sum of the first n cubes.

$$\left(\frac{n(n+1)}2\right)^2=\frac {n^2(n+1)^2}{4}.$$

I've looked at derivations of all the formulas and they make sense to me, but I'm curious about why my approach landed me where it did. Note, I'm currently taking Calculus 2.

amWhy
  • 209,954
Nono
  • 19
  • 1
    Try adding two non-trivial numbers. Now square the result. Is this the same as first squaring and then adding? If it doesn't work on just two things, why would you expect it to work on a large set of things? – Nij Mar 04 '22 at 19:19
  • 1
    Nice find, @RobPratt! Thanks. – amWhy Mar 04 '22 at 19:48
  • 1
    First, I would say that this is a very interesting line of thought, and I would strongly urge you to reconsider your belief that you are "not very good at [math]". Nij's comment is half of the answer: they explain well why your approach did not land you where you thought it was going to. But it is certainly surprising that your approach did land you anywhere at all, and you are right to be curious. It seems that the proofs you've read (and perhaps those you will read in the duplicate question) did not totally sate your curiosity.... – Eric Nathan Stucky Mar 04 '22 at 19:56
  • 1
    ... When this happens it is often because there is an even more interesting question that you need to formulate first. For me that question is "Just how lucky did you get?" e.g. if you go around squaring sums of $k^{\text{th}}$ powers (or perhaps cubing them, or...), do you often land on sums of $\ell^\text{th}$ powers? is this the only time this happens? or is the truth somewhere in between? // But this is only the question that is most interesting to me, and I give it only as an example of what such a followup question might look like. The important thing is to chase your own interest :) – Eric Nathan Stucky Mar 04 '22 at 19:58
  • @EricNathanStucky That's why I thought to ask instead of writing this off as yet another shot in the dark that didn't quite work out (because it kind of sort of did!) Thank you for your input! I now have a new question to chase :D – Nono Mar 04 '22 at 20:41

0 Answers0