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?

The formula $1+2+3+...+n$ mean the "sum of all integers starting with $1$ and going up to $n$". You can substitute any positive integer you want in place of $n$, for instance by substituting $n=6$ you get $1 + 2 + 3 + ... + 6$, which ordinarily we would write out more explicitly as $1+2+3+4+5+6$, and them sum to get $21$.
Similarly, the formula $1+2+3+....+BLAH$ to represent the "sum of all integers starting with $1$ and going up to BLAH. You can substitute any positive integer you want in place of BLAH.
Finally the formula $1+2+3+...+2n$ mean "the sum of all integers starting with $1$ and going up to $2n$". You can substitute any positive integer you want in place of $n$, and then do the arithmetic to get $2n$, and then substitute that amount. For instance, if you take $n=3$ and substitute that into the expression $2n$ then you get $2n = 2 \cdot 3 = 6$, and then when you substitute $2n=6$ into the formula $1+2+3+...+2n$ you get $1+2+3+...+6 = 21$.
