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Home » 8w = 320 \\ - RoadRUNNER Motorcycle Touring & Travel Magazine

8w = 320 \\ - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 April 29, 2026 👤 scraface

Apr 29, 2026

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📌 Take the positive root: \( n = \frac{30}{4} = 7.5 \). Since \( n \) must be an integer, the largest possible \( n \) is 7. Check \( S_7 \):

📌 S_7 = \frac{7}{2} (4(7) + 10) = \frac{7}{2} (38) = 7 \cdot 19 = 133 \leq 150

📌 \( S_8 = \frac{8}{2} (4(8) + 10) = 4 \cdot 42 = 168 > 150 \), so maximum is 7.

📌 Question: An ichthyologist observes that the population of a rare fish species in the Amazon doubles every 3 years. If the current population is 324 and it started at a power of 3, what is the smallest exponent \( k \) such that the population reaches at least \( 3^k \)?

📌 Solution: The population grows as \( 324 \times 2^{t/3} \), where \( t \) is time in years. We want the smallest \( k \) such that:

📌 \times 2^{t/3} \geq 3^k

📌 But since we are to express the population as a power of 3, note that 324 = \( 3^4 \times 4 = 81 \times 4 = 2^2 \cdot 3^4 \). So:

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