\(A=1-\left(\frac{3}{2.10}+\frac{3}{4.15}+\frac{3}{6.20}+....+\frac{3}{198.500}\right)\)

  \(=1-\frac{3}{2.5}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\right)\)

   \(=1-\frac{3}{25}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\right)\)

   \(=1-\frac{3}{25}\left(1-\frac{1}{100}\right)\)

\(=1-\frac{3.99}{25.100}=\frac{703}{1000}\)

\(1-\frac{3}{2.10}-\frac{3}{4.15}-\frac{3}{6.20}-\frac{3}{8.25}-...-\frac{3}{198.500}\)

\(=1-\left(\frac{3}{2.10}+\frac{3}{4.15}+\frac{3}{6.20}+...+\frac{3}{198.500}\right)\)

  \(=1-\frac{3}{2.5}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\right)\)

\(=1-\frac{3}{10}.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\right)\)

\(=1-\frac{3}{10}.\left(1-\frac{1}{100}\right)\)

\(=1-\frac{3}{10}.\frac{99}{100}\)

\(=1-\frac{297}{1000}\)

\(=\frac{703}{1000}\)

P/s : Không biết đúng hông nha, làm đại

B = 1 - 3/2×10 - 3/4×15 - 3/6×20 - ... - 3/198×500

B = 1 - (3/2×10 + 3/4×15 + 3/6×20 + ... + 3/198×500)

B = 1 - 3/2×5 × (1/1×2 + 1/2×3 + 1/3×4 + ... + 1/99×100)

B = 1 - 3/10 × (1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/99 - 1/100)

B = 1 - 3/10 × (1 - 1/100)

B = 1 - 3/10 × 99/100

B = 1 - 297/1000

B = 703/1000

\(A=\frac{3}{2.1.2.5}+\frac{3}{2.2.3.5}+\frac{3}{2.3.4.5}+\frac{3}{2.4.5.5}+...+\frac{3}{2.99.100.5}\)

\(A=\frac{3}{2.5}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\right)\)

Đặt biểu thức trong ngoặc đơn là B

\(B=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+\frac{5-4}{4.5}+...+\frac{100-99}{99.100}\)

\(B=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)

\(B=1-\frac{1}{100}=\frac{99}{100}\)

\(A=\frac{3}{2.5}.B=\frac{3.99}{2.5.100}=\frac{297}{1000}\)