The given sequence is:

1, 4, 7, 10, 13, 16

This is an arithmetic sequence with a constant difference of 3 between consecutive terms. The recurrence relation for this sequence can be expressed as:

Recurrence Formula

$a_n = a_{n-1} + 3 \quad \text{for } n \geq 2$ Where:

• $a_1 = 1$ (the first term of the sequence)
• $a_n$ represents the $n$-th term.

Explicit Formula

If you'd like the direct formula for the $n$-th term: $a_n = 1 + 3(n-1) \quad \text{or simply } \quad a_n = 3n - 2$

Would you like a detailed explanation or examples of how to use this? 😊

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Here are 5 related questions to explore further:

1. What is the 20th term in this sequence using the explicit formula?
2. How would the recurrence relation change if the difference between terms were 5 instead of 3?
3. Can you derive the explicit formula for other arithmetic sequences, e.g., $2, 5, 8, 11, 14$?
4. How do you find the sum of the first $n$ terms of an arithmetic sequence?
5. Can you derive a geometric sequence formula and compare it to the arithmetic formula?

Tip: Always verify by substituting a few terms into the recurrence formula!