<s>[INST] Arithmetic pattern $1 + 2 = 3$, $4 + 5 + 6 = 7 + 8$, and so on I am noticing this pattern:
\begin{align}
1+2&=3\\
4+5+6&=7+8\\
9+10+11+12&=13+14+15 \\
16+17+18+19+20&=21+22+23+24 \\
&\vdots
\end{align}
Is there a general formula that expresses a pattern here, such as for the $n$th term, or the closed form of the entire summation? I think the $n$th term starts with $$n^2+(n^2+1)+\cdots=\cdots+[(n+1)^2-1].$$
I am also presuming once this formula is discovered, we can prove it by induction for any $n$.
 [/INST] Another way to build up the pattern you see.
Start with the $n$th triangle number, $T_n$ (I'll use $n=5$):
$$1 + 2 + 3 + 4 + 5 = 1 + 2 + 3 + 4 + 5.$$
Add $n^2$ to both sides, except distribute the $n^2$ on one side by adding $n$ to each of the $n$ terms:
$$5^2 + 1 + 2 + 3 + 4 + 5 = 6 + 7 + 8 + 9 + 10.$$
Finally, add $n^3$ to both sides by distributing $n^2$ to all terms except the first one on the left:
$$25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35.$$
So, both sides are
$$T_n + n^2 + n^3 = \frac{n(n+1)}{2} + n^2 + n^3.$$
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