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Note: The last section has a repetition of the same URL, which is likely a mistake. The correct URL should be used only once. A sequence of integers starts with $1$, and each subsequent number is determined by adding to the previous number either $2$ or $3$. For example, starting from $1$, possible sequences could be $1, 3, 6, 9$ or $1, 4, 7, 10$. What is the smallest positive integer that can appear in this sequence exactly three times? To determine the smallest positive integer that can appear in the sequence exactly three times, we need to analyze the structure of the sequence. The sequence starts with (1) and each subsequent number is obtained by adding either (2) or (3) to the previous number. We will explore the possible values and their frequency. First, let's list out some initial terms of the sequence: - Starting from (1), we can add (2) or (3): - Adding (2): (1, 3) - Adding (3): (1, 4) Next, we continue from each of these starting points: - From (3): - Adding (2): (3, 5) - Adding (3): (3, 6) - From (4): - Adding (2): (4, 6) - Adding (3): (4, 7) From these steps, we see that the number (6) appears twice so far. Let's continue to explore further terms: - From (5): - Adding (2): (5, 7) - Adding (3): (5, 8) - From (7): - Adding (2): (7, 9) - Adding (3): (7, 10) - From (6) (again): - Adding (2): (6, 8) - Adding (3): (6, 9) Now, we need to check if any number appears exactly three times. Let's continue the exploration: - From (8): - Adding (2): (8, 10) - Adding (3): (8, 11) - From (9): - Adding (2): (9, 11) - Adding (3): (9, 12) We observe that the number (6) has appeared three times so far: - (1, 4, 7, 10, 12) (not including (6)) - (1, 4, 7, 10, 13) (not including (6)) - (1, 4, 7, 9, 12) (not including (6)) - (1, 4, 7, 9, 13) (not including (6)) To confirm, let's construct a sequence where (6) appears exactly three times: - Start with (1). - Add (2) to get (3). - Add (3) to get (6). - Add (2) to get (8). - Add (3) to get (11). Thus, the sequence is (1, 3, 6, 8, 11), and the number (6) appears exactly three times in this sequence. Therefore, the smallest positive integer that can appear in the sequence exactly three times is (oxed{6}).