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Home » 🎉 Unlock the Secret to a Life-Changing Happy Birthday Blessings You’ve Never Seen! - Midis

🎉 Unlock the Secret to a Life-Changing Happy Birthday Blessings You’ve Never Seen! - Midis

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 Reindex the sums:

📌 = \frac{1}{2} \left( \sum_{k=1}^{50} \frac{1}{k} \right) - \left( \sum_{k=2}^{51} \frac{1}{k} \right) + \frac{1}{2} \left( \sum_{k=3}^{52} \frac{1}{k} \right).

📌 Combine terms:

📌 = \frac{1}{2} \left( 1 + \frac{1}{2} + \sum_{k=3}^{50} \frac{1}{k} \right) - \left( \frac{1}{2} + \sum_{k=3}^{50} \frac{1}{k} + \frac{1}{51} \right) + \frac{1}{2} \left( \sum_{k=3}^{50} \frac{1}{k} + \frac{1}{51} + \frac{1}{52} \right).

📌 = \frac{1}{2} + \frac{1}{4} - \frac{1}{2} - \frac{1}{51} + \frac{1}{2} \cdot \frac{1}{51} + \frac{1}{2} \cdot \frac{1}{52} = \frac{1}{4} - \frac{1}{102} + \frac{1}{104}.

📌 \frac{1}{4} - \frac{1}{102} + \frac{1}{104} = \frac{1}{4} - \frac{52 - 51}{5304} = \frac{1}{4} - \frac{1}{5304} = \frac{1326 - 1}{5304} = \frac{1325}{5304}.

📌 Final answer: $\boxed{\dfrac{1325}{5304}}$Question:

📌 A micropaleontologist is studying a circular microfossil with a radius of 5 cm. If a concentric circular groove of width 1 cm is carved through the center, what is the area of the remaining microfossil material in square centimeters?

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