Given vectors $\displaystyle x_{1}, x_{2}, ..., x_{k}\in \mathbb{R}^{n}$, what is $\displaystyle |x_{1} + x_{2} + ... + x_{k}|^{2}$?

So $\displaystyle |x_{1} + x_{2}|^{2} = |x_{1}|^{2} + |x_{2}|^{2} + 2x_{1}\cdot x_{2}$ and extending it to the sum of three vectors it seems to be $\displaystyle |x_{1}|^{2} + |x_{2}|^{2} + |x_{3}|^{2} + 2( x_{1}\cdot x_{2} + x_{1}\cdot x_{3} + x_{2}\cdot x_{3})$. So it seems to be that for the norm of $\displaystyle k$ vectors we have $\displaystyle |x_{1} + x_{2} + ... + x_{k}|^{2} = |x_{1}|^{2} + |x_{2}|^{2} + ... + |x_{k}|^{2} + 2(\sum x_{i}x_{j})$ for $\displaystyle i\ne j, 1\le i, j \le k$. Is this correct and is there a better way to formulate this? And how would I go about giving a formal proof for this problem? Thanks.

Edit: I think I found a more simplified formulation: $\displaystyle \sum_{j=1}^{k}\sum_{i=1}^{k} x_{i}\cdot x_{j}$, but how would I go about giving a formal proof? Thanks.