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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.

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    If $a(n)=0$ for all $n$, then clealy $x_n=0$ for all $n$ – 2017-02-25
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    Yeah, I see what you mean; perhaps that wasn't the best example. I'll change it to a more suitable one. – 2017-02-25
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    The answer to the first question is, no. – 2017-02-25

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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.