The first of my questions may be entirely elementary, but the second (closely related) question may be of appropriate interest for this site.

Suppose that we are given $w_1, w_2, \cdots, w_n$ of positive integers which are co-prime. Let $H > 1$ be a positive parameter. I am interested in evaluation the sum

$$\displaystyle \sum_{w_1 x_1 + \cdots + w_n x_n \leq H} (w_1 x_1 + \cdots + w_n x_n)$$

I think the sum can be expressed in the form $c_0 H^{n+1} + O(H^{n})$ for some positive constant $c_0 > 0$, and if this is the case then I want to know what $c_0$ is in terms of the weights $w_1, \cdots, w_n$.

Second question is for general weights $w_1, \cdots, w_n$, where these are positive real numbers (but not necessarily integers), can we obtain the same result? Since a counting argument is not so clear here, what would be the way to show it?

Thanks for any insights.