every integers from 1 to 121 can be written as 5 powers of 3 We have a two-pan balance and have 5 integer weights with which it is possible to weight
exactly all the weights integers from 1 to 121 Kg.The weights can be placed all on a plate but you can also put some in a dish and others with the goods to be weighted.
It's asked to find the 5 weights that give us this possibility. It also asks you to prove that the group of five weights is the only one that solves the problem.
Easily i found the 5 weights: 1 , 3 , 9 , 27 , 81
but i can' demonstrate that this group of five weights is the only one that solves the problem.
Can you help me ?  
Thanks in advance !

A more general result says, given weights $w_1\le w_2 \le \dots \le w_n$, and if $S_k = \sum_{i=1}^k w_k$; $S_0 = 0$, then everything from $1$ to $S_n$ is weighable iff each of the following inequalities hold:
$$S_{k+1} \le 3S_k + 1 \text{ for } k = 0,\dots,n-1$$
Note that this is equivalent to $w_k \le 2 S_k +1$ for each $k$.
If you wish, a proof is given here. 
