Nhân cả tử cả mẫu của các phân số trong A với 2 ta có:

\(A=\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+..........+\frac{2}{90}\)

\(=2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+.........+\frac{1}{90}\right)\)

\(=2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+........+\frac{1}{9.10}\right)\)

\(=2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+......+\frac{1}{9}-\frac{1}{10}\right)\)

\(=2\left(\frac{1}{2}-\frac{1}{10}\right)\)

\(=2.\frac{2}{5}\)

\(=\frac{4}{5}\)

\(A=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+...+\frac{1}{45}\)

\(A=\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+\frac{2}{30}+...+\frac{2}{90}\)

\(A=2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{9.10}\right)\)

\(A=2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{9}-\frac{1}{10}\right)\)

\(A=2\left(\frac{1}{2}-\frac{1}{10}\right)\)

\(A=\frac{2}{5}\)

\(A=\frac{4}{5}\)

Lời giải:

$\frac{A}{2}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+\frac{1}{90}$
$=\frac{2-1}{1\times 2}+\frac{3-2}{2\times 3}+\frac{4-3}{3\times 4}+\frac{5-4}{4\times 5}+\frac{6-5}{5\times 6}+\frac{7-6}{6\times 7}+\frac{9-8}{8\times 9}+\frac{10-9}{9\times 10}$

$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}$

$=1-\frac{1}{9}=\frac{8}{9}$

$\Rightarrow A=2\times \frac{8}{9}=\frac{16}{9}$

\(A=\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+\dfrac{1}{15}+\dfrac{1}{21}+\dfrac{1}{28}+\dfrac{1}{36}+\dfrac{1}{45}+\dfrac{1}{55}\)

\(A=2\times\dfrac{1}{2}\times\left(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+\dfrac{1}{15}+\dfrac{1}{21}+\dfrac{1}{28}+\dfrac{1}{36}+\dfrac{1}{45}+\dfrac{1}{55}\right)\)

\(A=2\times\left(\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}+\dfrac{1}{56}+\dfrac{1}{72}+\dfrac{1}{90}+\dfrac{1}{110}\right)\)

\(A=2\times\left(\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+\dfrac{1}{4\times5}+\dfrac{1}{5\times6}+...+\dfrac{1}{9\times10}+\dfrac{1}{10\times11}\right)\)

\(A=2\times\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{10}-\dfrac{1}{11}\right)\)

\(A=2\times\left(\dfrac{1}{2}-\dfrac{1}{11}\right)\)

\(A=2\times\dfrac{9}{22}\)

\(A=\dfrac{9}{11}\)

(1 - 1/3) × (1 - 1/6) × (1 - 1/10) × (1 - 1/15) × ... × (1/780) × a = 1

2/3 × 5/6 × 9/10 × 14/15 × ... × 779/780 × a = 1

4/6 × 10/12 × 18/20 × 28/30 × ... × 1558/1560 × a = 1

1×4/2×3 × 2×5/3×4 × 3×6/4×5 × 4×7/5×6 × ... × 38×41/39×40 × a = 1

1×2×3×4×...×38/2×3×4×5×...×39 × 4×5×6×7×...×41/3×4×5×6×...×40 × a = 1

1/39 × 41/3 × a = 1

41/297 × a = 1

=> a = 297/41

1 - 1/3) × (1 - 1/6) × (1 - 1/10) × (1 - 1/15) × ... × (1/780) × a = 1

2/3 × 5/6 × 9/10 × 14/15 × ... × 779/780 × a = 1

4/6 × 10/12 × 18/20 × 28/30 × ... × 1558/1560 × a = 1

1×4/2×3 × 2×5/3×4 × 3×6/4×5 × 4×7/5×6 × ... × 38×41/39×40 × a = 1

1×2×3×4×...×38/2×3×4×5×...×39 × 4×5×6×7×...×41/3×4×5×6×...×40 × a = 1

1/39 × 41/3 × a = 1

41/297 × a = 1

=> a = 297/41

\(a=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+...+\frac{1}{45}\)

\(a=\frac{1}{1.3}+\frac{1}{2.3}+\frac{1}{2.5}+\frac{1}{3.5}+...+\frac{1}{5.9}\)

\(a=2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{9.10}\right)\)

\(a=2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\right)\)

\(a=2\left(\frac{1}{2}-\frac{1}{10}\right)\)

=> \(a=2.\frac{2}{5}\)

=> \(a=\frac{4}{5}\)

\(A=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+\frac{1}{28}+\frac{1}{45}\)

\(\Rightarrow\frac{1}{2}A=\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}\right)\cdot\frac{1}{2}\)

\(=\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+\frac{1}{90}\)

\(=\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\frac{1}{6\cdot7}+\frac{1}{7\cdot8}+\frac{1}{8\cdot9}+\frac{1}{9\cdot10}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}\)

\(=\frac{1}{2}-\frac{1}{10}=\frac{2}{5}\)

\(\Rightarrow A=\frac{2}{5}:\frac{1}{2}=\frac{4}{5}\)

\(a=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+....+\frac{1}{45}\)

\(a=\frac{1}{1.3}+\frac{1}{2.3}+\frac{1}{2.5}+\frac{1}{3.5}+...+\frac{1}{5.9}\)

\(\frac{1}{2}.a=\frac{1}{2}\left(\frac{1}{1.3}+\frac{1}{2.3}+\frac{1}{2.5}+\frac{1}{3.5}+...+\frac{1}{5.9}\right)\)

\(\frac{1}{2}.a=\frac{1}{2.1.3}+\frac{1}{2.2.3}+\frac{1}{2.2.5}+\frac{1}{2.3.5}+...+\frac{1}{2.5.9}\)

\(\frac{1}{2}.a=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{9.10}\)

\(\frac{1}{2}.a=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\)

\(\frac{1}{2}.a=\frac{1}{2}-\frac{1}{10}\)

\(\frac{1}{2}.a=\frac{2}{5}\)

\(a=\frac{2}{5}:\frac{1}{2}=\frac{2}{5}.2\)

=> \(a=\frac{4}{5}\)

= 7536573657865734657365873464876

\(=1-2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+......+\frac{1}{90}\right)=1-2\left(\frac{1}{2.3}+\frac{1}{3.4}+.....+\frac{1}{9.10}\right)=1-2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-.......-\frac{1}{10}\right)=1-2\left(\frac{1}{2}-\frac{1}{10}\right)=1-\frac{2.4}{10}=1-\frac{4}{5}=\frac{1}{5}\)