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Home » r = \sqrt{\frac50\pi} - DNSFLEX

r = \sqrt{\frac50\pi} - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \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:

📌 \( f(0) = c = 4 \)

📌 \( f(1) = a + b + c = 1 \) → \( a + b + 4 = 1 \) → \( a + b = -3 \)

📌 \( f(2) = 4a + 2b + c = -4 \) → \( 4a + 2b + 4 = -4 \) → \( 4a + 2b = -8 \) → \( 2a + b = -4 \)

📌 a + b = -3 \\

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with equality if and only if \( a = b = c \). Since \( a + b + c = 1 \...
\frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}}...
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 b...
\frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sq...
\sum_{n=1}^{50} \left( \sqrt{n+1} - \sqrt{n} \right) = (\sqrt{2} - \sq...

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