1. ## finite or infinite?

A set of numbers is defined as a1=1, a2=2, a3=3, and a number n belongs to this set if it can be written as a unique sum of three distinct members of this set e.g. a4=6, a5=9, a6=10, a7=11, a8=12 but 13, 14, 15, 16, 17 do not belong to this set. Is this set of numbers finite or infinite? Prove your claim.
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I've been trying to solve this question by contradiction. So I assume that the set is finite. I have no idea how to proceed. can anyone give me a hint?

2. I think contradiction is fine. Assume the set is finite, so it is bounded above. That means that your set, since now S, have a max say $a_\left\{max{{}\right\}=m$ then show that you can construct another number that belongs to the set and is greater than m

3. An additional observation is that if the set is finite, what will happen if you add the three largest numbers of that set. will the sum be in the set? Should it be in the set?

4. From the examples in the OP, $a=a_{n-2}+a_{n-1}+a_n$ may indeed belong to the set; however, it is not clear to me why $a$ is represented as a sum of three distinct members in a unique way.

5. Originally Posted by Isomorphism
An additional observation is that if the set is finite, what will happen if you add the three largest numbers of that set. will the sum be in the set? Should it be in the set?
I've tried working that out. It turns out that it isn't necessary that the sum of the three largest numbers will belong to set. Counter Example is 18.
Working:
Suppose S = {1,2,3,6,9} so far.
now 3+6+9 = 18, which should be in this set S.
BUT it is not because when we work out more elements of S, we get S = {1,2,3,6,9,10,11,12} and from here, we clearly see that 18 has another representation i.e. 11+6+1.
So 18 has two representations!

6. ## solved!

oh I think I've figured out the mistake I was making. Once you suppose that you're set is finite, you take 3 largest numbers and add them to get a number x. This x can either be in the set or must have another triple-number representation. But if it has a triple number representation then there must be a number bigger than the 3 largest numbers we took, leading to a contradiction!
thank you all for the help... I'm most grateful