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Home » Did You See the Dawn of The Amazing Spider-Man 2? It’s Mind-Blowing! - Dyverse

Did You See the Dawn of The Amazing Spider-Man 2? It’s Mind-Blowing! - Dyverse

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 So the sum is \( 3 \cdot \frac{1}{2} = \frac{3}{2} \).

📌 Thus, the minimum value is \( \boxed{\frac{3}{2}} \).

📌 Question: Compute \( \sum_{n=1}^{50} \frac{1}{\sqrt{n} + \sqrt{n+1}} \).

📌 Solution: Consider the general term:

📌 \frac{1}{\sqrt{n} + \sqrt{n+1}}

📌 Rationalize the denominator by multiplying numerator and denominator by \( \sqrt{n+1} - \sqrt{n} \):

📌 \frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = \frac{\sqrt{n+1} - \sqrt{n}}{(n+1) - n} = \sqrt{n+1} - \sqrt{n}

📌 \sum_{n=1}^{50} \left( \sqrt{n+1} - \sqrt{n} \right) = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + \cdots + (\sqrt{51} - \sqrt{50})

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