Number addition riddle I got this math "riddle" in one of my math test, and I would love to know how to solve it.
If
$$S = 1 + 2 + 3 + 4 + \ldots + 2015,$$
then a sum of
$$1 + 2 + 3 + \ldots + 2015 + 2016 + \ldots + 4030$$
is equal to: ?
I know the answer is  $2S + 2015^2$, but not how to get it. Could someone explain, please?

$$
\begin{align}
&1+2+\cdots+2015+2016+\cdots+4030\\
=\:&S+(2015+1)+(2015+2)+\cdots+(2015+2015)\\
=\:&S+(1+2+\cdots+2015)+(\underbrace{2015+2015+\cdots+2015}_{2015\text{ times}})\\
=\:&S+S+2015\cdot2015\\
=\:&2S+2015^2
\end{align}
$$
If you want to find the value of $S$, you can do the following additional pairing:
$$
\begin{align}
S&=1+2+\cdots+2015\\
&=(2015+1)+(2014+2)+\cdots+(1009+1007)+1008\\
&=\underbrace{2016+2016+\cdots+2016}_{1007\text{ times}}+1008\\
&=1007\cdot2016+1008\\
&=2\,031\,120
\end{align}
$$
