Problem 6

Let

a_1,a_2,a_3,\ldots

be a sequence of positive real numbers. Suppose that for some positive integer

s

, we have

a_n=\max\{a_k+a_{n-k}\mid 1\le k\le n-1\}

for all

n>s

. Prove that there exist positive integers

\ell

and

N

, with

\ell\le s

, such that

a_n=a_\ell+a_{n-\ell}

for all

n\ge N

Step 1 of 9: Start from the max recurrence

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a_n=\max_{1\le k\le n-1}(a_k+a_{n-k})\quad(n>s)

Detailed analysis

The hypothesis is $a_n=\max_{1\le k\le n-1}(a_k+a_{n-k})\quad(n>s)$ . Whenever a summand on the right has an index greater than $s$ , expand that term again using the same rule.