Sequence of $a_n=(-\frac{1}{2})^n$ Considering the sequence of $a_n=(-\frac{1}{2})^n$ where n is a natural number, I have to prove that it is a sequence in which the positive terms and negative terms generate two geometric illiminated progressions having the same ratio then calculate the sum of the terms of $a_n$.
The final part is easy but I don't know how to give a demonstration of what the exercise is asking. I can find some terms then take the positive one and the negative one and finally prove that the two sequences I found are two geometric progression having the same ratio... But I find not very elegant... Can you help me please?
Thanks in advance!
(following my solution the ratio is 1/4) 

Note that, for every $n\geqslant0$,
$$a_{2n}=r^n,\qquad a_{2n+1}=b\cdot r^n,
$$
with
$$
b=-\frac12,\qquad r=\frac14,$$
hence
$$
\sum_{n=0}^\infty a_{2n}=\frac1{1-r}=\frac43\qquad\sum_{n=0}^\infty a_{2n+1}=\frac{b}{1-r}=-\frac23,
$$
and
$$
\sum_{n=0}^\infty a_{n}=\frac43-\frac23=\frac23.
$$
