To write the explicit formula for the sequence defined by the recursive formula:

$b(1) = b_1 \quad \text{and} \quad b(n) = b(n-1) - 6$

Step-by-step process:

1. Understand the sequence behavior: The recursive formula states that each term is 6 less than the previous term. Therefore, the sequence decreases by 6 with each step.

2. Find the general term: Let's write out the first few terms:

   • $b(1) = b_1$
   • $b(2) = b(1) - 6 = b_1 - 6$
   • $b(3) = b(2) - 6 = b_1 - 6 - 6 = b_1 - 2 \cdot 6$
   • $b(4) = b(3) - 6 = b_1 - 3 \cdot 6$

   From this pattern, you can generalize the formula: $b(n) = b_1 - (n-1) \cdot 6$

3. Simplify the formula: $b(n) = b_1 - 6(n-1)$

4. Final explicit formula: $b(n) = b_1 - 6n + 6$

   Alternatively, it can be written as: $b(n) = 6 + b_1 - 6n$

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Explicit formula:

$b(n) = b_1 - 6n + 6$

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Would you like a deeper explanation of how this formula was derived? Let me know!

Related questions:

1. How would you find the explicit formula if the recursive step was an addition instead of subtraction?
2. What happens if the first term $b_1$ is a fraction or a decimal?
3. Can this formula be used to find negative terms in the sequence (e.g., $n = -1$)?
4. How would the sequence behave if the decrement were a variable, such as $k$, instead of $6$?
5. How can you prove the explicit formula is correct using mathematical induction?

Tip:

When working with sequences, always verify your explicit formula by plugging in small values of $n$ to ensure it matches the recursive formula!