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30 tháng 10 2018

\(A=\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right).....+\left(1+\frac{1}{99.101}\right)\)

\(=\frac{1.3+1}{1.3}.\frac{2.4+1}{2.4}.\frac{3.5+1}{3.5}.....\frac{99.101+1}{99.101}\)

\(=\frac{2^2}{1.3}.\frac{3^2}{2.4}.\frac{4^2}{3.5}.....\frac{100^2}{99.101}\)

\(=\frac{2.3.4.....100}{1.2.3.....99}.\frac{2.3.4.....100}{3.4.5.....101}\)

\(=100.\frac{2}{101}=\frac{200}{101}\)

11 tháng 5 2016

\(\left(1+\frac{1}{1.3}\right).\left(1+\frac{1}{2.4}\right).\left(\frac{1}{3.5.}\right).....\left(1+\frac{1}{99.101}\right)\)

\(=\frac{4}{3}.\frac{9}{8}.\frac{16}{15}.....\frac{10000}{9999}\)

\(=\frac{2^2}{1.3}.\frac{3^2}{2.4}.\frac{4^2}{3.5}.....\frac{100^2}{99.101}\)

\(=\frac{2^2.3^2.4^2.5^2.....98^2.99^2.100^2}{1.2.3^2.4^2.5^2......99^2.100.101}\)

\(=\frac{2.100}{1.101}\)

\(=\frac{200}{101}\)

11 tháng 4 2016

1-1/101=100/101

nha bạn         

11 tháng 4 2018

\(\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)...\left(1+\frac{1}{99.101}\right)\)

\(=\frac{4}{1.3}.\frac{9}{2.4}....\frac{10000}{99.101}\)

\(=\frac{2.2.3.3...100.100}{1.3.2.4...99.101}\)

\(=\frac{\left(2.3.4...100\right)\left(2.3.4...100\right)}{\left(1.2...99\right)\left(3.4.5...101\right)}\)

\(=\frac{100.2}{101}=\frac{200}{101}\)

12 tháng 4 2021

\(D=\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right)...\left(1+\frac{1}{99.101}\right)\)

\(D=\frac{4}{1.3}.\frac{9}{2.4}.\frac{16}{3.5}...\frac{10000}{99.101}\)

\(D=\frac{2^2}{1.3}.\frac{3^2}{2.4}.\frac{4^2}{3.5}...\frac{100^2}{99.101}\)

\(D=\frac{2.3.4...100}{1.2.3...99}.\frac{2.3.4...100}{3.4.5...101}=100.\frac{2}{101}=\frac{200}{101}\)

Vậy  \(D=\frac{200}{101}\)

6 tháng 7 2017

= 4/1.3 x 9/2.4 x 16/3.5 x...x 10000/99.101

= 2.2/1.3 x 3.3/2.4 x 4.4/3.5 x..x 100.100/99.101

= (2.3.4. ... 100/1.2.3. .... 99) x (2.3.4. ... .100/3.4.5. ... .101)

= 100.2/101

=200/101

7 tháng 3 2018

\(A=\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right)...\left(1+\frac{1}{99.101}\right)\)

\(\Rightarrow A=\frac{1.3+1}{1.3}.\frac{2.4+1}{2.4}.\frac{3.5+1}{3.5}.....\frac{99.101+1}{99.101}\)

\(\Rightarrow A=\frac{4}{1.3}.\frac{9}{2.4}.\frac{16}{3.5}.....\frac{10000}{99.101}\)

\(\Rightarrow A=\frac{2^2}{1.3}.\frac{3^2}{2.4}.\frac{4^2}{3.5}.....\frac{100^2}{99.101}\)

\(\Rightarrow A=\frac{\left(2.3.4.....100\right)\left(2.3.4.....100\right)}{\left(1.2.3.....99\right)\left(3.4.5.....101\right)}\)

\(\Rightarrow A=\frac{100.2}{101}=\frac{200}{101}\)

6 tháng 7 2017

\(A=\left(1+\dfrac{1}{1.3}\right)\left(1+\dfrac{1}{2.4}\right)\left(1+\dfrac{1}{3.5}\right)....\left(1+\dfrac{1}{99.101}\right)\)

\(A=\dfrac{4}{1.3}.\dfrac{9}{2.4}.\dfrac{16}{3.5}....\dfrac{10000}{99.101}\)

\(A=\dfrac{2^2}{1.3}.\dfrac{3^2}{2.4}.\dfrac{4^2}{3.5}....\dfrac{100^2}{99.101}\)

\(A=\dfrac{2.3.4...100}{1.2.3....99}.\dfrac{2.3.4....100}{3.4.5....101}\)

\(A=100.\dfrac{2}{101}=\dfrac{200}{101}\)

Vậy A = \(\dfrac{200}{101}\)

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