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Home » This is a telescoping sum. Write out the first few terms: - Abbey Badges

This is a telescoping sum. Write out the first few terms: - Abbey Badges

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 9e^{-0.5t} = 1

📌 e^{-0.5t} = \frac{1}{9}

📌 -0.5t = \ln\left(\frac{1}{9}\right) = -\ln 9

📌 Solve for \(t\):

📌 t = \frac{-\ln 9}{-0.5} = \frac{\ln 9}{0.5} = 2 \ln 9 = \ln(9^2) = \ln 81

📌 Alternatively, express the final result clearly:

📌 t = \frac{\ln 9}{0.5} = 2 \ln 9 = 2 \ln(3^2) = 4 \ln 3

📌 Thus, the time when the population reaches 500 is:

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