<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] Let $n = 4$ and $m = 2n = 8$.
Then $1 + 2 + 3 + 4 + 5  +6+7+8 = 36$.
$1+2+3+4+5+6+7+8 = \frac {8(8+1)}{2}= \frac {2*4(2*4 + 1)}2 = 36$.
$1 + 2 + 3 + ....... + m = \frac {m(m+1)}{2}$ and if $m = 8$ then $1 + 2 + 3 +....... + 8 = \frac {8(8+1)}{2} = 36$.
And $1+2+3+...... + 2n -1 + 2n = \frac {2n(2n+1)}2$ and if $n= 4$ then $1 + 2+ 3 + ...... + 8 = \frac {(2*4)(2*4 + 1) } 2 = 36$.
It's nothing profound or difficult.
$1 + 2 + 3 +.......... + whatever = \frac {whatever(whatever + 1)}2$.
And $whatever$ can be .... whatever you want.
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