MathDB

Let

A

be a finite set of positive integers, and for each positive integer

n

we define

S_n = \{x_1 + x_2 + \cdots + x_n \;\vert\; x_i \in A \text{ for } i = 1, 2, \dots, n\}

That is,

S_n

is the set of all positive integers which can be expressed as sum of exactly

n

elements of

A

, not necessarily different. Prove that there exist positive integers

N

and

k

such that

\left\vert S_{n + 1} \right\vert = \left\vert S_n \right\vert + k \text{ for all } n\geq N.

Proposed by Ariel García

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