Question

[nguyenthibao an]

Cho \(A=\)\(\frac{1}{1\cdot21}+\frac{1}{2\cdot22}+\frac{1}{3\cdot23}+...+\frac{1}{80.100}\)                                                                                                                                         \(B=\frac{1}{1\cdot81}+\frac{1}{2\cdot82}+\frac{1}{3\cdot83}+...+\frac{1}{20\cdot100}\)        
Tính \(\frac{A}{B}\)                                                 

Answer 1

[四种草药 - TFBoys]

A=1/1.21 + 1/ 2.22 +1/3.23 +......+ 1/80.100

A=1/20 .(20/1.21 +20/2.22 + 20/3.23 +....+ 20/80.100)

A=1/20 .(1-1/21 +1/2 -1/22 +....+1/80 - 1/100)Aa= 1/20 . ( 1+1/2 =1/3 + ....+1/80 )-(1/21 +1/22 +....+ 1/100 )

A= 1/20 .(1+1/2 +.....+1/20 -1/81 -....-1/100)Bb=11.81 +1/2.82 +....+ 1/20.100

B=1/1.81 +1/ 2.82 +1/ 3.83 +....+ 1/ 20.100

B=1/80 .(80/1.81 +80/2.82 +....+ 80/20.100)

B=1/80.(1-1/81 +1/2 -1/82 +....+ 1/20 -1/100

B=A/B =1/20:1/80 =4

VẬY A/B =4

Answer 2

[zZz Cool Kid_new zZz]

\(A=\frac{1}{1\cdot21}+\frac{1}{2\cdot22}+\frac{1}{3\cdot23}+....+\frac{1}{80\cdot100}\)

\(20A=\frac{1}{1}-\frac{1}{21}+\frac{1}{2}-\frac{1}{22}+\frac{1}{3}-\frac{1}{23}+...+\frac{1}{80}-\frac{1}{100}\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{20}\right)+\left(\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+\frac{1}{24}+....+\frac{1}{80}\right)-\left(\frac{1}{21}+\frac{1}{23}+....+\frac{1}{80}\right)-\left(\frac{1}{81}+\frac{1}{82}+...+\frac{1}{99}+\frac{1}{100}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{20}\right)-\left(\frac{1}{81}+\frac{1}{82}+...+\frac{1}{99}+\frac{1}{100}\right)\left(1\right)\)

\(B=\frac{1}{1\cdot81}+\frac{1}{2\cdot82}+\frac{1}{3\cdot83}+....+\frac{1}{20\cdot100}\)

\(80B=\frac{1}{1}-\frac{1}{81}+\frac{1}{2}-\frac{1}{82}+\frac{1}{3}-\frac{1}{83}+....+\frac{1}{20}-\frac{1}{100}\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{20}\right)-\left(\frac{1}{81}+\frac{1}{82}+\frac{1}{83}+....+\frac{1}{100}\right)\left(2\right)\)

Từ \(\left(1\right);\left(2\right)\) \(\Rightarrow\frac{A}{B}=1\)