Question

\(A=\frac{1}{1x21}+\frac{1}{2x22}+\frac{1}{3x23}+...+\frac{1}{80x100}\)

\(B=\frac{1}{1x81}+\frac{1}{2x82}+...+\frac{1}{20x100}\)

TÍNH \(\frac{A}{B}\)

Answer 1

\(Tacocongthuc:\frac{1}{n}-\frac{1}{n+k}=\frac{k}{n\left(n+k\right)}\)

\(A=\frac{1}{1.21}+\frac{1}{2.22}+...+\frac{1}{80.100}\Rightarrow20A=\frac{20}{1.21}+\frac{20}{2.22}+.....+\frac{20}{80.100}=1-\frac{1}{21}+\frac{1}{2}-\frac{1}{22}+.....+\frac{1}{80}-\frac{1}{100}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{20}-\frac{1}{81}-\frac{1}{82}-....-\frac{1}{100}\)

\(B=\frac{1}{1.81}+\frac{1}{2.82}+.....+\frac{1}{20.100}\Rightarrow80B=\frac{80}{1.81}+\frac{80}{2.82}+....+\frac{80}{20.100}=1-\frac{1}{81}+\frac{1}{2}-\frac{1}{82}-....+\frac{1}{20}-\frac{1}{100}=1+\frac{1}{2}+\frac{1}{3}+.......+\frac{1}{20}-\frac{1}{81}-\frac{1}{82}-.....-\frac{1}{100}\)

\(\Rightarrow20A=80B\Leftrightarrow A=4B\Rightarrow\frac{A}{B}=4\)

Answer 2

sao 1-1/21+1/2-1/22+1/3-1/23+...+1/80-1/100 lại bằng 1+1/2+1/3+...+1/20-1/81-1/82-...-1/100         đúng ra phải là 1+1/2+1/3+...+1/80-1/21-1/22-1/23-...-1/100