<s>[INST] How do I interpret this sum? So if the sum of $n$ integers $\ge 1$ equal $\frac{n(n+1)}2$. Then my book goes on and says $1 + 2 + 3 +\ldots + 2n = \frac{2n(2n + 1) }2$. 
I'm confused about what $1 + 2 + 3 + \ldots +2n$ means. If the sequence is $1, 2, 3, 4$ then where does $2n$ have to do with the $n$th number?
 [/INST] Compare:
$$\color{red}{1+2+\cdots +n}=\frac{n(n+1)}{2}$$
and
$$\color{red}{1+2+\cdots+n}+\color{blue}{(n+1)+(n+2)+\cdots+2n}=$$
$$\color{red}{1+2+\cdots+n}+\color{blue}{(1+2+\cdots+n)+n\cdot n}=$$
$$\color{red}{\frac{n(n+1)}{2}}+\color{blue}{\frac{n(n+1)}{2}+n^2}=\frac{2n(2n+1)}{2}$$
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