Question:
The least positive integer \(n\) for which \(n!<\left(\frac{n+1}{2}\right)^n\) holds is
The least positive integer \(n\) for which \(n!<\left(\frac{n+1}{2}\right)^n\) holds is
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Test small values of \(n\) sequentially until the inequality becomes true.
Updated On: Apr 23, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Formula / Definition}
\[ n!<\left(\frac{n+1}{2}\right)^n \]
Step 2: Calculation / Simplification}
\(n=1: 1! = 1; \left(\frac{2}{2}\right)^1 = 1 \Rightarrow 1<1\) (False)
\(n=2: 2! = 2; \left(\frac{3}{2}\right)^2 = 2.25 \Rightarrow 2<2.25\) (True)
\(n=3: 3! = 6; \left(\frac{4}{2}\right)^3 = 8 \Rightarrow 6<8\) (True)
For all \(n \geq 2\), the inequality holds.
Step 3: Final Answer
\[ n = 2 \]
\[ n!<\left(\frac{n+1}{2}\right)^n \]
Step 2: Calculation / Simplification}
\(n=1: 1! = 1; \left(\frac{2}{2}\right)^1 = 1 \Rightarrow 1<1\) (False)
\(n=2: 2! = 2; \left(\frac{3}{2}\right)^2 = 2.25 \Rightarrow 2<2.25\) (True)
\(n=3: 3! = 6; \left(\frac{4}{2}\right)^3 = 8 \Rightarrow 6<8\) (True)
For all \(n \geq 2\), the inequality holds.
Step 3: Final Answer
\[ n = 2 \]
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