Originally Posted by

**craig** Prove by induction that, for all $\displaystyle n \subset N$:

$\displaystyle 1^3 + 3^3 + 5^3 + ... + (2n - 1)^3 = n^2(2n^2 - 1).$

Prove for $\displaystyle n=1$:

LHS, $\displaystyle 1^3 = 1$

RHS, $\displaystyle 1^2(2-1)=1$

True for $\displaystyle n=1$, now assume for $\displaystyle n=k$,

$\displaystyle 1^3 + 3^3 + 5^3 + ... + (2k - 1)^3 = k^2(2k^2 - 1)$

Now the sequence above is for add numbers, assuming that $\displaystyle k$ is odd, then instead of using $\displaystyle n=k+1$ use $\displaystyle n=k+2$.

We then have:

$\displaystyle 1^3 + 3^3 + 5^3 + ... + (2(k+2) - 1)^2$

$\displaystyle 1^3 + 3^3 + 5^3 + ... + (2k+4 - 1)^2$

$\displaystyle 1^3 + 3^3 + 5^3 + ... + (2k+3)^2$.

Not sure where to go from here? Any help would be great.

Thank you