Re: Square of sum = sum of cubes Do you have any explaination why? If you were to derive the summation formula, you'd get

sum (n) = (n)(n+1)/2

And with

sum (n^3) = ((n)(n+1)/{2})^2

So you see its true for all n. But, I don't see why. Anyone else have any insights.

Re: Square of sum = sum of cubes

you can prove it by mathematical induction... since it's very late here, i think i'll post the proof tomorrow

Re: Square of sum = sum of cubes Assume: (1 + 2 +…+ n)^2 = 1^3 + 2^3 + … + n^3

*Exploration
(1 + 2)^2 = 9 = 1^3 + 2^3
(1 + 2 + 3)^2 = 36 = 1^3 + 2^3 + 3^3
(1 + 2 + 3 + 4)^2 = 100 = 1^3 + 2^3 + 3^3 + 4^3
(1 + 2 + 3 + 4 + 5)^2 = 225 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3
*Recall
Pythagoras’ triangular number Tn = 1 + 2 + 3 +…+ n
Solve for Tn in terms of n:
Sol.
Let Tn=S:
S = 1 + 2 + 3 + … + (n-1) + n
S + S = 2S = (1+n) + (2+(n-1)) + … + ((n-1)+2) + (n+1))
2S= n(n+1)
S = Tn = n(n+1)/2
This can be proven true by induction for any n
*Substitution
Therefore, observing our original statement,
(Tn)2 = 1^3 + 2^3 + … + n^3
*Simplify the cubes
*Exploration
1^3 + 2^3 = 9 = (2 + 1)^2
1^3 + 2^3 + 3^3 = 36 = (3 + 3)^2
1^3 + 2^3 + 3^3 + 4^3 = 100 = (4 + 6)^2
1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 225 = (5 + 10)^2
A pattern emerges:
1^3 + 2^3 + … + n^3 = (n + Tn-1)^2, or
1^3 + 2^3 + … + n^3 = (n + (n-1)(n)/2)^2, or
1^3 + 2^3 + … + n^3 = ( (n^2+n)/2)^2, or simply
1^3 + 2^3 + … + n^3 = (Tn)^2
*Substitution
(Tn)^2 = (Tn)^2

This is a very round-about proof, but it shows that the assumption holds true, so the square of a sum of the first n natural numbers is equal to the sum of their cubes

proof by induction

I'm not so sure this proof by induction is valid, it seems that simply a pattern was observed, but no actual induction took place, i.e. there is no demonstration that the nth term implies the n+1 term to be true. What is shown above is inductive reasoning, not inductive proof.

I am working on a proof myself, if i finish it i will post, I would also appreciate any comments regarding the validity of the inductive proof shown. Inductive proof is relatively new to me, last year in discrete we covered it but only briefly.