Question

How would I find the sum of the first 31 cubes?

Answer 1

There is a beautiful explanation of this without words here: http://books.google.com/books?id=cyyhZr-SffcC&lpg=PA67&dq=proofs%20without%20words%20sum%20of%20cubes&pg=PA85#v=onepage&q&f=false

Answer 2

I cant look at that and understand it, it means nothing to me.

Answer 3

that's beautiful

Answer 4

Oh! Epic sauce look what I found!
1 8 27 64 125
7 19 37 61
12 18 24 - Multiples of 6!
So something like (6x1 + 6x2 + 6x3 + 6x4...)
which is 6(1 + 2 + 3 + 4... + n)!
So with my theory the answer should be 6(1 + 2 +... + 31)
so 6 x 496 is definitely not the answer :(
way too small
I think it has something to do with how for the first 5 cubes, i only had 3 multiples of 6, maybe multiply that by n + 2, so 33?
That equals 98,208, which is still way too small. The smallest possible answer is 216,225. Help please?

Answer 5

the formula is printed very clearly on mathteacher's link\[\large1^3+2^3+3^3+\dots+n^3=(1+2+3+\dots+n)^2\]

Answer 6

So it would be 496^2?

Answer 7

So say it was the first 31 fourth-powers, it would be 496^3?

Answer 8

But could you help me with my theory that I was trying to establish?

Answer 9

Look at the formula for the sum if the first n integers raise to power 1, 2, and 3 respectively:\[\sum_{i=1}^{n}i=\frac{n(n+1)}{2}\]\[\sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1)}{6}\]\[\sum_{i=1}^{n}i^3=\frac{n^2(n+1)^2}{4}=(\sum_{i=1}^{n}i)^2\]There is almost a pattern here, but not really. I don't know of any formulas for higher power series, let alone patterns among them all. I'm not sure such patterns are even fully known or understood.

Answer 10

oh look, I just found this!
http://www.mathpages.com/home/kmath279/kmath279.htm
so I guess there is a pattern after all :D

Answer 11

Okay, I'm not sure what the greek letter means, but I think I get the gist of them. Thanks for your time. Could you help me out with my theory though?

Answer 12

oh, great! I'll take a look at it.

Answer 13

I wish I could generalize this stuff, but I'm not that smart. Looks like somebody was though. Enjoy, I'm reading too.

Answer 14

I can't figure that out. I don't understand what it means. :( Now I'm frustrated...

Answer 15

The greek letter is sigma. It just means add up all the numbers from the bottom to the top part\[\sum_{i=n}^{5}n=1+2+3+4+5\]for instance, or\[\sum_{n=1}^{3}n^2=1^2+2^2+3^2\]and yes, I never said that the pattern would be obvious. Good luck deciphering it.

Answer 16

typo, first series should be\[\sum_{n=1}^{5}n=1+2+3+4+5\]

Answer 17

It's too small for me to read anyways, but thanks for trying.

Answer 18

I couldnt even begin to understand that link.

Answer 19

the sum of square is annoying, but the sum of cubes is actually nicer, since it is just the square of the sum of consecutive natural numbers

Answer 20

yes.

Answer 21

there is a way of deriving each of them from the previous one, but the algebra gets increasingly more painful