To find the explicit formula for the sequence $b(n)$, we start with the recursive formula:

$b(n) = b(n-1) - 6$ and the initial condition: $b(1) = 8$

Step 1: Expand the recursive formula

Substitute $b(n-1)$ in terms of $b(n-2)$: $b(n) = b(n-2) - 6 - 6 = b(n-2) - 12$ Substituting further: $b(n) = b(n-3) - 6 - 6 - 6 = b(n-3) - 18$ Generalizing this: $b(n) = b(1) - 6(n-1)$

Step 2: Use the initial condition

Substitute $b(1) = 8$: $b(n) = 8 - 6(n-1)$

Step 3: Simplify

Simplify the formula: $b(n) = 8 - 6n + 6$ $b(n) = 14 - 6n$

Final Explicit Formula

$b(n) = 14 - 6n$

Would you like a step-by-step example or additional clarification?

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Questions to Explore:

1. What are the first five terms of the sequence using the explicit formula?
2. How can the explicit formula be used to calculate $b(10)$?
3. What does the sequence represent graphically (e.g., linear decrease)?
4. How is this sequence different from a geometric sequence?
5. Can the recursive formula be converted back from the explicit formula?

Tip:

When working with recursive sequences, deriving an explicit formula allows for direct computation of terms without iteration!