Cứu với!!!!!!!!!!!!!

Đáp án + Giải thích các bước giải:

$\begin{align} \textbf{A} &= \dfrac{1}{1 \cdot 21} + \dfrac{1}{2 \cdot 22} + \dfrac{1}{3 \cdot 23} + \ldots + \dfrac{1}{80 \cdot 100} 
\\ 20\textbf{A} &= \dfrac{20}{1 \cdot 21} + \dfrac{20}{2 \cdot 22} + \dfrac{20}{3 \cdot 23} + \ldots + \dfrac{20}{80 \cdot 100} 
\\ &= 1 - \dfrac{1}{21} + \dfrac{1}{2} - \dfrac{1}{22} + \dfrac{1}{3} - \dfrac{1}{23} +\ldots + \dfrac{1}{80} - \dfrac{1}{100}
\\ &= 1 + \dfrac{1}{2} +\dfrac{1}{3} + \ldots + \dfrac{1}{20} + \bigg(\dfrac{1}{21} - \dfrac{1}{21}\bigg) + \bigg(\dfrac{1}{22} - \dfrac{1}{22}\bigg) + \ldots +
\\ & \bigg(\dfrac{1}{80} - \dfrac{1}{80}\bigg) - \dfrac{1}{81} - \dfrac{1}{82} - \ldots - \dfrac{1}{100}
\\ &= 1 + \dfrac{1}{2} + \dfrac{1}{3}+ \ldots + \dfrac{1}{20} - \dfrac{1}{81} -\dfrac{1}{82} - \ldots - \dfrac{1}{100}\end{align}$

$\begin{align} \textbf{B} &= \dfrac{1}{1 \cdot 81} + \dfrac{1}{2 \cdot 82} + \dfrac{1}{3 \cdot 83} + \ldots + \dfrac{1}{20 \cdot 100} 
\\ 80\textbf{B} &= \dfrac{80}{1 \cdot 81} + \dfrac{80}{2 \cdot 82} + \dfrac{80}{3 \cdot 83} + \ldots + \dfrac{80}{20 \cdot 100} 
\\ &= 1 - \dfrac{1}{81} + \dfrac{1}{2} - \dfrac{1}{82} + \dfrac{1}{3} - \dfrac{1}{83} +\ldots + \dfrac{1}{20} - \dfrac{1}{100}
\\ &= 1 + \dfrac{1}{2} + \dfrac{1}{3}+ \ldots + \dfrac{1}{20} - \dfrac{1}{81} -\dfrac{1}{82} - \ldots - \dfrac{1}{100}\end{align}$

Suy ra $20\textbf{A} = 80\textbf{B}$

$\textbf{A} = 4\textbf{B}$

$\dfrac{\textbf{A}}{\textbf{B}} = 4$