\(1\dfrac{1}{3};1\dfrac{1}{8};1\dfrac{1}{15};...\\ \Leftrightarrow\dfrac{4}{3};\dfrac{9}{8};\dfrac{16}{15};...\\ \Leftrightarrow\dfrac{2^2}{1\cdot3};\dfrac{3^2}{2\cdot4};\dfrac{4^2}{3\cdot5};...\)
98 số hạng đầu tiên của dãy là:
\(\dfrac{2^2}{1\cdot3};\dfrac{3^2}{2\cdot4};\dfrac{4^2}{3\cdot5};...;\dfrac{99^2}{98\cdot100}\)
Tích của chúng là:
\(\dfrac{2^2}{1\cdot3}\cdot\dfrac{3^2}{2\cdot4}\cdot\dfrac{4^2}{3\cdot5}\cdot...\cdot\dfrac{99^2}{98\cdot100}\\ =\dfrac{2\cdot2}{1\cdot3}\cdot\dfrac{3\cdot3}{2\cdot4}\cdot\dfrac{4\cdot4}{3\cdot5}\cdot...\cdot\dfrac{99\cdot99}{98\cdot100}\\ =\dfrac{2\cdot2\cdot3\cdot3\cdot4\cdot4\cdot...\cdot99\cdot99}{1\cdot3\cdot2\cdot4\cdot3\cdot5\cdot...\cdot98\cdot100}\\ =\dfrac{\left(2\cdot3\cdot4\cdot...\cdot99\right)\cdot\left(2\cdot3\cdot4\cdot...\cdot99\right)}{\left(1\cdot2\cdot3\cdot...\cdot98\right)\cdot\left(3\cdot4\cdot5\cdot...\cdot100\right)}\\ =\dfrac{2\cdot3\cdot4\cdot...\cdot99}{1\cdot2\cdot3\cdot...\cdot98}\cdot\dfrac{2\cdot3\cdot4\cdot...\cdot99}{3\cdot4\cdot5\cdot...\cdot100}\\ =\dfrac{99}{1}\cdot\dfrac{2}{100}\\ =99\cdot\dfrac{1}{50}\\ =\dfrac{99}{50}\)
