Question

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Tính nhanh:

a, M=3/1*2+3/2*3+3/3*4+........+3/20*21

b, N=4/3*9/8*16/15*25/24*.......*100/99

Answer 1

[Mashiro Shiina]

\(M=\dfrac{3}{1.2}+\dfrac{3}{2.3}+\dfrac{3}{3.4}+...+\dfrac{3}{20.21}\)

\(M=3\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{20}-\dfrac{1}{21}\right)\)

\(M=3\left(1-\dfrac{1}{21}\right)\)

\(M=3.\dfrac{20}{21}=\dfrac{20}{7}\)

\(N=\dfrac{4}{3}.\dfrac{9}{8}.\dfrac{16}{15}.\dfrac{25}{24}...\dfrac{100}{99}\)

\(N=\dfrac{4.9.16.25...100}{3.8.15.24...99}\)

\(N=\dfrac{2.2.3.3.4.4.5.5...10.10}{1.3.2.4.3.5.4.6...9.11}\)

\(N=\dfrac{2.3.4.5...10}{1.2.3...9}.\dfrac{2.3.4.5...10}{3.4.5...11}\)

\(N=10.\dfrac{2}{11}=\dfrac{20}{11}\)

Answer 2

[Nguyễn Khánh Huyền]

a) \(M=\dfrac{3}{1.2}+\dfrac{3}{2.3}+\dfrac{3}{3.4}+......+\dfrac{3}{20.21}\)

= \(3.\left(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+......+\dfrac{1}{20}-\dfrac{1}{21}\right)\)

= \(3.\left(\dfrac{1}{1}-\dfrac{1}{21}\right)\)

= \(3.\dfrac{20}{21}\)

= \(\dfrac{20}{7}\)

b) \(N=\dfrac{4}{3}.\dfrac{9}{8}.\dfrac{16}{15}.\dfrac{25}{24}.......\dfrac{100}{99}\)

= \(\dfrac{4.9.16.25.....100}{3.8.15.24.....99}\)

= \(\dfrac{2.2.3.3.4.4.5.5.......10.10}{1.3.2.4.3.5.4.6......9.11}\)

= \(\dfrac{\left(2.3.4.5.....10\right).\left(2.3.4.5.....10\right)}{\left(1.2.3.4......9\right).\left(3.4.5.....11\right)}\)

= \(\dfrac{10.2}{1.11}\)

= \(\dfrac{20}{11}\)