I've been practicing induction and I came across this problem:
Consider the following series, 1, 2, 3, 4, 5, 10,20, 40, ..., which starts as an arithmetic series, but after the first 5 terms becomes a geometric series. Prove that any positive integer can be written as a sum of distinct numbers from this series.
I've proved a similar problem to this concerning adding 4 and 5 cent stamps to come up with the correct postage. However, that problem doesn't stipulate that I use distinct numbers, so it is straightforward to prove with the well-ordering principle.
Can anyone help me figure out a way to handle this problem?