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Xét tử : \(1.3.5.....99\)
\(=\frac{1.2.3.4.....98.99.100}{2.4.6.....100}\)
\(=\frac{\left(1.2.3.....50\right)\left(51.52.....99.100\right)}{\left(1.2\right).\left(2.2\right).....\left(50.2\right)}\)
\(=\frac{\left(1.2.3.....50.\right).\left(51.52.....100\right)}{\left(1.2.3.....50\right).2.2.....2}\)
\(=\frac{51.52.....100}{2.2....2}\)
\(=\frac{51}{2}.\frac{52}{2}.....\frac{100}{2}\)
Ta được phân số\(\frac{\frac{51}{2}.\frac{52}{2}.....\frac{100}{2}}{51.52.....100}\)
\(=\frac{\frac{51}{2}.\frac{52}{2}.....\frac{100}{2}}{\frac{51}{2}.\frac{52}{2}.....\frac{100}{2}.2.2.....2}\)
\(=\frac{1}{2.2.....2}\)
\(=\frac{1}{2^{50}}\)
\(\frac{51.52.53...100}{1.3.5...99}\)
\(=\frac{\left(2.4.6...100\right).\left(51.52.53...100\right)}{\left(2.4.6...100\right).\left(1.3.5...99\right)}\)
\(=\frac{\left(2.4.6...100\right).\left(51.52.53...100\right)}{1.2.3.4.5.6...99.100}\)
\(=\frac{2.4.6...100}{1.2.3...50}\)
\(=\frac{\left(2.2...2\right).\left(1.2.3...50\right)}{1.2.3...50}\)
\(=2.2.2...2\)
\(=2^{50}\)