induction - How do I express a series where the first term is positive and all subsequent terms are negative?

Let's say I have a series $a(n)$ like this:
$$a(1)=x_1$$
$$a(2)=3x_1-x_2$$
$$a(3)=5x_1-3x_2-x_3$$
$$a(4)=7x_1-5x_2-3x_3-x_4$$
$$\vdots$$
I can express $a(n)$ as follows, if I'm not mistaken:
$$a(n)=(2n-1)x_1-\sum_{i=1}^{n-1}{(2i-1)x_{n-i+1}}$$
However, I find this way of expressing it extremely unwieldy and, quite simply, ugly. Is there a better way to express such a series?

Also, given that $a(n)+f(n)=0$ for some function $f$, where all $n\in \mathbb{Z}^+$, is it possible to prove this statement via mathematical induction? I have tried my hand at it, but I find that, due to the fact that only the first term is positive, I am unable to do so.

If not possible through induction, is there another method I can use?

Apologies if the question or title is confusing or misleading, and for asking two questions in one post.

Answer

As far as the relation you've given, what you have written is correct. It is unfortunate that the relation is not simpler.

However, another relation can be derived. $a_1=x_1$, $a_2=3x_1-x_2$, and for $n\ge3$,
$$

a_n-2a_{n-1}+a_{n-2}=-x_n-x_{n-1}
$$
This might be easier to use.