proof by induction for summation This is not a duplicate.  I know my conjecture is right, just need to prove it, using induction(not Gauss Method).
Conjecture formula from following equations, and prove conjecture:
$1=1,\\2+3+4=1+8,\\5+6+7+8+9=8+27,\\10+11+12+13+14+15+16=27+64\\$

$S(n)=\sum_{i=(n-1)^2+1}^{n^2}i=(n-1)^3+n^3$

$S(1)=1=1\\S(2)=2+3+4=1+8\\S(3)=5+6+7+8+9=8+27\\S(4)=10+11+12+13+14+15+16=27+64$

Need to show $S(n+1)=\sum_{i=n^2+1}^{(n+1)^2}i=n^3+(n+1)^3$

It's kind of hairy with the variable in the summation.  Seems like there should be a simpler way to prove it than trying to work out that summation.

Assume $S(n)$ true.  So $S(n+1)=(n-1)^3+n^3+?$

Thanks.

Yes, I am aware that $\sum_{i=k}^ni=\sum_{i=1}^ni-\sum_{i=1}^{k-1}i$ and that $\sum_{i=1}^ni = \frac{n(n+1)}{2}$ and that can be used.
Use $S(n+1)=\sum_{i=1}^{(n+1)^2}i-(n-1)^3+(n)^3$
Use $S(n+1)=\sum_{i=1}^{(n+1)^2}i-(n-1)^3+(n)^3$