8 Positive integers making numbers in the range $-1985\leq k\leq 1985$.

Question

Problem. (IMO 1985, Longlisted Problem 25). Find $8$ positive integers $n_1, \ldots, n_8$ with the following property. For every integer $k$, $-1985\leq k\leq 1985$, there are $8$ integers $\alpha_1, \ldots, \alpha_8$, each belonging to $\{-1, 0, 1\}$, such that $k = \sum_{i=1}^8 \alpha_i n_i$.

In other words, we need to find $8$ positive integers, such that adding or subtracting them yields every integers in the interval [-1985, 1985].

Answer 1

HINT: Let $n_k = 3^k$; $k=0,1,2,\ldots, 7$.

Answer 2

I used the HINT by Mike.
Mike, thank you very much for your HINT.

$\{a_03^0+\dots+a_73^7\mid a_i\in\{0,1,2\}\}=\{0,1,\dots,6560\}.$
So,
$\{a_03^0+\dots+a_73^7-(1\cdot 3^0+\dots+1\cdot 3^7)\mid a_i\in\{0,1,2\}\}=\{b_03^0+\dots+b_73^7\mid b_i\in\{-1,0,1\}\}=\{-3280,\dots,3280\}\supset\{-1985,\dots,1985\}.$