3/4 . 8/9 . 15/16 ... 9999/10000
= 1.3/2.2 . 2.4/3.3 ... 99.101/100.100
= 1 . 2 . ... . 99 / 2 . 3 . 100 × 3 . 4 ... 101 / 2 . 3 ... 100
= 1 / 100 . 101 / 2
= 101 / 200
=1.3/2.2 .2.4/3.3 .3.5/4.4 . ...... 99.101/100.100
=1.2.3.4.5 ...... .99/2.3.4.....100 . 3.4.5 ....... .101/2.3.4.5 .... .100
=1/100 .101/2
=101/200
k cho mink nha
\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...\frac{9999}{10000}\)
\(=\frac{3.8.15...9999}{4.9.16...10000}\)
\(=\frac{\left(1.3\right).\left(2.4\right).\left(3.5\right)...\left(99.101\right)}{\left(2.2\right).\left(3.3\right).\left(4.4\right)...\left(100.100\right)}\)
\(=\frac{\left(1.2.3...99\right).\left(3.4.5...101\right)}{\left(2.3.4...100\right).\left(2.3.4...100\right)}\)
\(=\frac{1.101}{100.2}\)
\(=\frac{101}{200}\)
Ta có: \(\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{9999}{10000}\)
\(=\frac{1\cdot3}{2\cdot2}\cdot\frac{2\cdot4}{3\cdot3}\cdot\frac{3\cdot5}{4\cdot4}\cdot...\cdot\frac{99\cdot101}{100\cdot100}\)
\(=\frac{1\cdot3\cdot2\cdot4\cdot3\cdot5\cdot...\cdot99\cdot101}{2\cdot2\cdot3\cdot3\cdot4\cdot4\cdot...\cdot100\cdot100}\)
\(=\frac{\left(1\cdot2\cdot3\cdot...\cdot99\right)\cdot\left(3\cdot4\cdot5\cdot...\cdot101\right)}{\left(2\cdot3\cdot4\cdot...\cdot100\right)\cdot\left(2\cdot3\cdot4\cdot...\cdot100\right)}\)
\(=\frac{1\cdot101}{100\cdot2}\)
\(=\frac{101}{200}\)
1*3/2*2.2*4/3*3.3*5/4*4.....99*101/100*100. =1*2*3*...*99/2*3*4*...*100.3*4*5*...*101/2*3*4*...*100. =1/100 . 101/2. =101/200
