understanding a signal from its mathematical formula
What is misleading is that you ought to find the values of $x_1[n]$ for all $n$. 

For the first term, assuming convention $\cdot^0=1$, you get $\delta[n]+\delta[n-2]$. For the second one, all terms except for $k=2$ cancel each other, so you only gain another $\delta[n-2]$. Then, as you guessed, the last term (without the initial minus sign), is 1 for $n=3,4$, and zero elsewhere.