a: \(A=\frac89\cdot\frac{15}{16}\cdot\frac{24}{25}\cdot\frac{35}{36}\cdot\frac{48}{49}\cdot\frac{63}{64}\)
\(=\frac{2\cdot4}{3\cdot3}\cdot\frac{3\cdot5}{4\cdot4}\cdot\frac{4\cdot6}{5\cdot5}\cdot\frac{5\cdot7}{6\cdot6}\cdot\frac{6\cdot8}{7\cdot7}\cdot\frac{7\cdot9}{8\cdot8}\)
\(=\frac{2\cdot3\cdot4\cdot5\cdot6\cdot7}{3\cdot4\cdot5\cdot6\cdot7\cdot8}\cdot\frac{4\cdot5\cdot6\cdot7\cdot8\cdot9}{3\cdot4\cdot5\cdot6\cdot7\cdot8}=\frac28\cdot\frac93=\frac14\cdot3=\frac34\)
=>Nghịch đảo của A là \(\frac{1}{A}=\frac43\)
Ta có công thức tổng quát:
\(1-\frac{1}{\frac{n\left(n+1\right)}{2}}\)
\(=1-\frac{2}{n\left(n+1\right)}\)
\(=\frac{n\left(n+1\right)-2}{n\left(n+1\right)}=\frac{n^2+n-2}{n\left(n+1\right)}=\frac{\left(n+2\right)\left(n-1\right)}{n\left(n+1\right)}\)
\(B=\left(1-\frac13\right)\left(1-\frac16\right)\cdot\ldots\cdot\left(1-\frac{1}{28}\right)\)
\(=\left(1-\frac26\right)\left(1-\frac{2}{12}\right)\cdot\ldots\cdot\left(1-\frac{2}{56}\right)\)
\(=\left(1-\frac{2}{2\cdot3}\right)\left(1-\frac{2}{3\cdot4}\right)\cdot.\ldots\cdot\left(1-\frac{2}{7\cdot8}\right)\)
\(=\frac{\left(2+2\right)\left(2-1\right)}{2\left(2+1\right)}\cdot\frac{\left(3+2\right)\left(3-1\right)}{3\left(3+1\right)}\cdot\ldots\cdot\frac{\left(7+2\right)\left(7-1\right)}{7\cdot\left(7+1\right)}\)
\(=\frac{4\cdot5\cdot\ldots\cdot9}{3\cdot4\cdot\ldots\cdot8}\cdot\frac{1\cdot2\cdot\ldots\cdot6}{2\cdot3\cdot\ldots\cdot7}=\frac93\cdot\frac17=\frac37\)
=>Nghịch đảo của B là \(\frac{1}{B}=\frac73\)
b: \(\frac{1}{A}+\frac{1}{B}=\frac43+\frac73=\frac{11}{3}\)
