Question

[Phạm Gia Khánh]

\(\left(\frac{3}{1.8}+\frac{3}{8.15}+\frac{3}{15.22}+...+\frac{3}{106.113}\right)-\left(\frac{25}{50.55}+\frac{25}{55.60}+...+\frac{25}{95.100}\right)\)

Giải giúp mk với!

Answer 1

[Đoàn Đức Hà]

\(A=\frac{3}{1.8}+\frac{3}{8.15}+\frac{3}{15.22}+...+\frac{3}{106.113}\)

\(=\frac{3}{7}\left(\frac{7}{1.8}+\frac{7}{8.15}+\frac{7}{15.22}+...+\frac{7}{106.113}\right)\)

\(=\frac{3}{7}\left(\frac{8-1}{1.8}+\frac{15-8}{8.15}+\frac{22-15}{15.22}+...+\frac{113-106}{106.113}\right)\)

\(=\frac{3}{7}\left(1-\frac{1}{8}+\frac{1}{8}-\frac{1}{15}+\frac{1}{15}-\frac{1}{22}+\frac{1}{106}-\frac{1}{113}\right)\)

\(=\frac{3}{7}\left(1-\frac{1}{113}\right)=\frac{48}{113}\)

\(B=\frac{25}{50.55}+\frac{25}{55.60}+...+\frac{25}{95.100}\)

\(=5\left(\frac{5}{50.55}+\frac{5}{55.60}+...+\frac{5}{95.100}\right)\)

\(=5\left(\frac{55-50}{50.55}+\frac{60-55}{55.60}+...+\frac{100-95}{95.100}\right)\)

\(=5\left(\frac{1}{50}-\frac{1}{55}+\frac{1}{55}-\frac{1}{60}+...+\frac{1}{95}-\frac{1}{100}\right)\)

\(=5\left(\frac{1}{50}-\frac{1}{100}\right)=\frac{1}{20}\)

Giá trị của biểu thức đã cho là: 

\(A-B=\frac{48}{113}-\frac{1}{20}=\frac{847}{2260}\)