Please write an equation to define the sequence 1,3,7,9,13,18,21,25,30,35,41,47

The sequence you provided is: \(1, 3, 7, 9, 13, 18, 21, 25, 30, 35, 41, 47\).

This sequence doesn't follow a single common difference, so it's not arithmetic. However, we can observe the differences between consecutive terms to find a pattern:

- \(3 - 1 = 2\)
- \(7 - 3 = 4\)
- \(9 - 7 = 2\)
- \(13 - 9 = 4\)
- \(18 - 13 = 5\)
- \(21 - 18 = 3\)
- \(25 - 21 = 4\)
- \(30 - 25 = 5\)
- \(35 - 30 = 5\)
- \(41 - 35 = 6\)
- \(47 - 41 = 6\)

Now, let's try to define the sequence by recognizing a pattern in the differences. It appears that the sequence might alternate or vary in its increments. 

A possible recursive formula for the sequence might be:

\[ a_n = a_{n-1} + d_n \]

Where:
- \( a_1 = 1 \)
- \( d_2, d_4, d_6, d_8, d_{10}, d_{12} \) alternate between 2, 4, and 5.
- \( d_3, d_5, d_7, d_9, d_{11} \) have values 4, 5, and 6 as seen in the differences above.

This makes the sequence somewhat complex to describe with a simple equation. However, it appears the sequence could be expressed as a combination of arithmetic-like progressions with alternating or varying steps.

To express it non-recursively, we could split the sequence into two or more sequences where each one has a simpler pattern, but the initial recursive approach captures the variability of the sequence directly.

Would you like more details on this or explore different ways to express this sequence?

Here are 5 related questions you might find interesting:
1. What are the general forms of arithmetic sequences?
2. How do you derive the formula for the sum of a sequence?
3. How can you determine if a sequence is arithmetic or geometric?
4. What are recursive formulas and how are they used?
5. How can a sequence be split into multiple sub-sequences for easier analysis?

**Tip:** Recursive formulas are powerful for sequences that do not follow a simple linear or geometric pattern, as they allow each term to be defined in terms of the previous one.

Discover how to define the sequence 1, 3, 7, 9, 13, 18, 21, 25, 30, 35, 41, 47 using a recursive formula. This problem explores the mathematical concepts of sequences, patterns, and recursive formulas, making it suitable for students in Grades 9-12.

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