Can you prove the summation 13+23+33+...+n3=(1+2+...n)2?

[Loren Booda]

Given nonzero whole numbers n, prove

1³+2³+3³+...+n³=(1+2+...n)²

I figured this out numerically, but lack the skills to solve it analytically (no doubt by induction) and could not find it in my table of summations. I'm too old for this to be homework.

 

[ice109]

Given nonzero whole numbers n, prove

1³+2³+3³+...+n³=(1+2+...n)²

I figured this out numerically, but lack the skills to solve it analytically (no doubt by induction) and could not find it in my table of summations. I'm too old for this to be homework.

http://en.wikipedia.org/wiki/Faulhaber's_formula

look at the case p=3

 

[K.J.Healey]

also
http://en.wikipedia.org/wiki/Squared_triangular_number

 

[Loren Booda]

Thanks, ice109. I'm surprised that I've never heard of Faulhaber before. I guess Bernoulli got all the acclaim.

I still don't see a derivation of my finding, however. I thought in the case I presented that p=3 on one side of the equation, and p=2 on the other, as opposed to p=3 for both sides of Faulhaber's formula.

K.J.Healey seems to have what I seek, including a proof of Nicomachus's theorem.

 

[ice109]

i like that one better

what the hell? every n^3 is the sum of n consecutive odd numbers? wheatstone's proof seems to imply that

 

[Gib Z]

If one knows the closed forms for all cases p< n, then the closed form for p=n can be derived as such:

Set up a table into two columns, LHS and RHS of the following equation;

(x+1)^{n+1} - x^{n+1} = (^{n+1}C_1)x^n + (^{n+1}C_2)x^{n-1} ...+1.

Sum this expression for k=1, 2,3,4...m. The LHS is a telescoping series. The RHS is the sum of cases p=0, 1, 2, 3... n. Replace every series with its known closed form. Then isolate the p=n case onto one side of the equation, and simplify.

Since we only need this for up to p=3, it shouldn't be very hard.