Let $V$ be a vector space over a field $K$ and let $v_1, ... v_n \in V$ be linearly independent. Show that $v_1$, $v_1 + v_2$, $\ldots$, $v_1 + v_2 + \cdots+ v_n$ are linearly independent.
This question here Suppose $v_1,v_1+v_2,v_1+v_2+v_3,...,\sum_{i=1}^nv_i$ are linear independent. Show that $v_1,v_2,...,v_n$ are also linear independent. seems to do the opposite.
I can answer my own question, thanks to @ArturoMagidin and @RyszardSzwarc
If they are linearly independent, then ($a_1 + ... a_n = 0$), ($a_2 + ... a_n = 0$), ... ($a_n = 0$). Because if they are linearly independent, then the only solution to $b_1v_1 + ... b_nv_n = 0$ is $b_1 = ... = b_n = 0$. So in our case $b_1 = a_1 + ... a_n, ..., b_n = a_n$. So $b_n = a_n = 0$. We have $a_n + a_{n-1} = 0$, we know $a_n = 0 \implies a_{n-1} = 0$. If we do that for all equations, we get $a_n = a_{n-1} = ... a_1 = 0$.