Prove the theorem of Nicomachus by induction Prove the theorem of Nicomachus(AD.100) by induction:
$$ 
1^3 = 1,\  2^3 = 3+ 5,\  3^3 = 7 + 9 + 11,\  4^3 = 13 + 15 + 17 + 19,\  ...
$$
My approach: from looking at the above pattern you can tell there is something of the following sort:
 $$ 2^{n-1} + q = n ^3,$$
where $q$ is odd s.t. $q = 2k + 1.$

If induction is not mandatory,
let the $r$th term$(T_r)$ of $1,3,7,13$  be $$a(r-1)(r-2)+b(r-1)+c$$
$r=1\implies 1=c$
$r=2\implies3=b+c\iff b=2$
$r=3\implies7=2a+2b+c\iff a=1$
$$T_r=(r-1)(r-2)+2(r-1)+1=r^2-r+1$$
Sum of $n$ term $$\sum_{m=\frac{n^2-n+1}2-1}^{\frac{n^2+n-1}2-1}(2m+1)=?$$
