Sequence where no addition two some elements yields another element

Question

I am trying to find a way to create a finite sequence of natural numbers with the property that no single element is the sum of a combination of other elements. It's simple with short sequences, is it possible as well if I have a minimum length of say 1000 elements?

So trying to put this into a formula; for a sequence of length n, each element $x_i$ and natural number constant $c_i \in [0, 1]$, the sum $\sum_{j=1}^{i-1} c_j*x_j + \sum_{j=i+1}^{n-1} c_j*x_j$ does not equal $x_i$.

Answer 1

If we make our sequence strictly increasing, we need only avoid $x_n=\sum_{j=1}^{n-1}c_jx_j$. That suggests to simply let $x_n=1+\sum_{j=1}^{n-1}x_j$. In fact, we can let $x_n=2^{n-1}$, i.e., the infinie sequence $1,2,4,8,16,\ldots$ works