To generate a closed formula for the sequence $5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, \dots$, we notice a pattern in the sequence:

• The sequence increases by 1, but every number (except the first one) appears twice. For example: $6$ appears twice, $8$ appears twice, $10$ appears twice, and so on.

This suggests that the sequence alternates between repeating terms and increasing by 1.

Finding the closed formula:

Define $a_n$ as the $n$-th term of the sequence. The sequence seems to be defined by an alternating pattern:

$a_n = 5 + \left\lfloor \frac{n-1}{2} \right\rfloor$

Where $\lfloor x \rfloor$ denotes the floor function, which rounds down to the nearest integer.

This formula works because:

• The floor function ensures that every integer $k$ appears twice. For odd $n$, it gives the same value as for the previous $n$, and for even $n$, it increments by 1.

Next four terms:

To find the next four terms of the sequence, we start from $n = 13$ since $a_{12} = 12$.

$a_{13} = 5 + \left\lfloor \frac{13-1}{2} \right\rfloor = 5 + \lfloor 6 \rfloor = 11$ $a_{14} = 5 + \left\lfloor \frac{14-1}{2} \right\rfloor = 5 + \lfloor 6.5 \rfloor = 11$ $a_{15} = 5 + \left\lfloor \frac{15-1}{2} \right\rfloor = 5 + \lfloor 7 \rfloor = 12$ $a_{16} = 5 + \left\lfloor \frac{16-1}{2} \right\rfloor = 5 + \lfloor 7.5 \rfloor = 12$

Thus, the next four terms are:

$11, 11, 12, 12$

Summary:

• Closed formula: $a_n = 5 + \left\lfloor \frac{n-1}{2} \right\rfloor$
• Next four terms: $11, 11, 12, 12$

Would you like further clarification or a deeper breakdown of the floor function?

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Here are 5 questions to expand on the topic:

1. How does the floor function help in generating sequences?
2. Can you generalize this sequence for any starting number?
3. What is the difference between floor and ceiling functions?
4. How would this sequence change if the numbers appear three times instead of twice?
5. Can you generate a recursive relation for this sequence instead of a closed formula?

Tip: When solving sequence problems, looking for patterns in how terms repeat or increase can help identify a general formula.