\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{899}{900}\)
\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}....\frac{29.31}{30.30}\)
\(=\frac{1.2.3....29}{2.3.4....30}.\frac{3.4.5....31}{2.3.4....30}\)
\(=\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)
\(A=\frac{1\cdot3}{2\cdot2}\cdot\frac{2\cdot4}{3\cdot3}\cdot...\cdot\frac{29\cdot31}{30\cdot30}\)
\(A=\frac{1\cdot2\cdot...\cdot29}{2\cdot3\cdot...\cdot30}\cdot\frac{3\cdot4\cdot...\cdot31}{2\cdot3\cdot...\cdot30}=\frac{1}{30}\cdot\frac{31}{2}=\frac{31}{60}\)
Ta có : \(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.....\frac{899}{900}\)
\(A=\frac{3.8.15.....899}{4.9.16.....900}\)
\(A=\frac{\left(1.3\right).\left(2.4\right).\left(3.5\right).....\left(29.31\right)}{2^2.3^2.4^2.....30^2}\)
\(A=\frac{\left(1.2.3.....29\right).\left(3.4.5.....31\right)}{\left(2.3.4.....30\right).\left(2.3.4.....30\right)}\)
\(A=\frac{1.31}{30.2}\)
\(A=\frac{31}{60}\)
A=\(\frac{1.3}{2^2}.\frac{2.4}{3^2}.....\frac{29.31}{30^2}\)
=\(\frac{1.3.2.4.....29.31}{2^2.3^2.....30^2}\)
=\(\frac{\left(1.2.3.....29\right).\left(3.4.....30.31\right)}{\left(2.3.....29.30\right).\left(2.3.4.....30\right)}\)
=\(\frac{1.31}{30.2}\)=\(\frac{31}{60}\)
\(\frac{1.3}{2.2}\)+ \(\frac{2.4}{3.3}\) + \(\frac{3.5}{4.4}\) + \(\frac{4.6}{5.5}\) + ..... + \(\frac{29.31}{30.30}\)
= \(\frac{1.2.3.4.....29}{2.3.4.5...30}\)+ \(\frac{3.4.5....31}{2.3.4.5...30}\)
=\(\frac{1}{3}\)+ \(\frac{31}{2}\)
= \(\frac{95}{6}\)