\(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+...+\frac{1}{97\cdot98\cdot99}+\frac{1}{98\cdot99\cdot100}\)
\(=\left(\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+...+\frac{2}{98\cdot99\cdot100}\cdot\frac{1}{2}\right)\)
\(=\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{98\cdot99}-\frac{1}{99\cdot100}\right)\cdot\frac{1}{2}\)
\(=\left(\frac{1}{2}-\frac{1}{99\cdot100}\right)\cdot\frac{1}{2}\)
\(=\frac{4949}{9900}\cdot\frac{1}{2}\)
\(=\frac{4949}{19800}\)
Đặt biểu thức trên là A, ta có
A=1/1x2x3+...+1/98x99x100
=> 2A=2/1x2x3+2/2x3x4+2/3x4x5...+2/98x99x100
=> 2A = -1/1x2+1/2x3-1/2x3+1/3x4-1/3x4-...-1/98x99+1/99x100
=> 2A = -1/1x2+1/99x100
=> 2A= -98/99
=> A= -98/198
Chúc bạn học tốt !
