To determine which formula is correct for the sequence \(24, 16, 8, 0, \ldots\), we can analyze the given terms.
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Identify the pattern in the given sequence: \[ \begin{align*} f(1) & = 24 \ f(2) & = 16 \ f(3) & = 8 \ f(4) & = 0 \ \end{align*} \]
We see that the sequence is decreasing by \(8\) each time.
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Find the general term: The first term is \(f(1) = 24\), the second term \(f(2) = 16\) can be expressed as: \[ f(2) = f(1) - 8 = 24 - 8 \cdot 1 \] For \(f(3)\): \[ f(3) = f(2) - 8 = 16 - 8 = 24 - 8 \cdot 2 \] For \(f(4)\): \[ f(4) = f(3) - 8 = 8 - 8 = 24 - 8 \cdot 3 \] From these calculations, we can see that the pattern can be expressed as: \[ f(n) = 24 - 8(n-1) \text{ (correcting to start with n=1)} \] Simplifying gives: \[ f(n) = 24 - 8n + 8 = 32 - 8n \]
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Get explicit forms from Felipe and Ling:
- Felipe: \(f(n) = 32 - 8n\)
- Ling: \(f(n) = 24 - 8n\)
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Check if each formula generates the sequence:
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For Felipe's formula \(f(n) = 32 - 8n\): \[ \begin{align*} f(1) & = 32 - 8(1) = 32 - 8 = 24 \ f(2) & = 32 - 8(2) = 32 - 16 = 16 \ f(3) & = 32 - 8(3) = 32 - 24 = 8 \ f(4) & = 32 - 8(4) = 32 - 32 = 0 \ \end{align*} \]
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For Ling's formula \(f(n) = 24 - 8n\): \[ \begin{align*} f(1) & = 24 - 8(1) = 24 - 8 = 16 \quad \text{(not correct!)} \ f(2) & = 24 - 8(2) = 24 - 16 = 8 \quad \text{(not correct!)} \ f(3) & = 24 - 8(3) = 24 - 24 = 0 \quad \text{(not correct!)} \ \end{align*} \]
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Based on this analysis, Felipe's formula \(f(n) = 32 - 8n\) is correct, while Ling's formula does not generate the correct sequence.