If $v_1,v_2,\dots,v_n$ are linearly independent then what about $v_1+v_2,v_2+v_3,\dots,v_{n-1}+v_n,v_n+v_1$? If $v_{1},v_{2},...v_{n}$ are linearly independent then are the following too?
$v_{1}+v_{2},v_{2}+v_{3},...v_{n-1}+v_{n},v_{n}+v_{1}$
I tried summing them to give 0 but had no success.

$$a_1(v_1+v_2) + a_2(v_2+v_3) + \cdots + a_{n-1}(v_{n-1}+v_n) + a_n(v_n + v_1) \\ = a_1v_1 + a_1v_2 + a_2v_2 + a_2v_3 + a_3v_3 + \cdots + a_{n-1}v_n + a_nv_n + a_nv_1 \\ = (a_1+a_n)v_1 + (a_1+a_2)v_2 + \cdots (a_{n-2} +a_{n-1})v_{n-1}+(a_{n-1}+a_n)v_n \\ = b_1v_1 + b_2v_2 + \cdots + b_{n-1}v_{n-1}+b_nv_n$$
Now make conclusions (casewise).
