Editorial for C bài 5.E2: Biểu thức phân số
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Lời giải chi tiết
Ta xét biểu thức \(\frac{1}{u \cdot v}\):
\(\frac{1}{u \cdot v} = \frac{1}{v - u} \cdot \frac{v-u}{u \cdot v} = \frac{1}{v-u}(\frac{1}{u} - \frac{1}{v})\)
Với biểu thức \(f(n, k)\), ta có:
\(f(n, k) = \frac{1}{k}(\frac{k}{1 \cdot (k+1)} + \frac{(2k+1)-(k+1)}{(k+1)(2k+1)}...\frac{(nk+1) - [(n-1)k+1]}{[(n-1)k+1] \cdot (nk+1)})\)
\(f(n, k) = \frac{1}{k}(1-\frac{1}{k+1}+\frac{1}{k+1}-\frac{1}{2k+1}+...+\frac{1}{(n-1)k+1}-\frac{1}{nk+1})\)
\(f(n, k) = \frac{1}{k}(1-\frac{1}{nk+1}) = \frac{1}{k} \frac{nk}{nk+1} = \frac{n}{nk+1}\)
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