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Home » Find a common denominator (12): - Simpleprint

Find a common denominator (12): - Simpleprint

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 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})

📌 \sqrt{51} - \sqrt{1} = \sqrt{51} - 1

📌 Therefore, the sum is \( \boxed{\sqrt{51} - 1} \).

📌 Question: Find the quadratic polynomial \( f(x) \) such that \( f(0) = 4 \), \( f(1) = 1 \), and \( f(2) = -4 \).

📌 Solution: Let \( f(x) = ax^2 + bx + c \). Use the given conditions:

📌 \( f(0) = c = 4 \)

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