HMMT 2001 V2

Find $\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right)$ in terms of $n$ .

Dataset text NuminaMath olympiads. Quoted with source.

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(-1)^{n-1}

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