\(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+....+\frac{1}{99\times100}\)
\(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(\frac{1}{1}-\frac{1}{100}\)
\(\frac{100-1}{100}\)
\(\frac{99}{100}\)
\(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{99\times100}\)
\(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(\frac{1}{1}-\frac{1}{100}\)
\(\frac{100-1}{100}\)
\(\frac{99}{100}\)
\(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{98\cdot99}+\frac{1}{99\cdot100}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)
\(=\frac{1}{1}-\frac{1}{100}\)
\(=\frac{100}{100}-\frac{1}{100}\)
\(=\frac{99}{100}\)
Tính:
1/( 1 x 2 ) + 1/( 2 x 3 ) + 1/( 3 x 4 ) + 1/( 4x 5 ) + ..... +1/( 98 x 99 ) + 1/( 99 x 100 )
= 2 - 1/ 1 x 2 + 3 - 2/2 x 3 + 4 -3/3 x 4 + 5 - 4/4 x 5 + ...... + 99 - 98/98 x 99 + 100 - 99/99 x 100
= 2/1 x 2 - 1/ 1 x 2 + 3/2 x 3 - 2/2 x 3 + 5/4 x 5 - 4/4 x 5 + ...... + 99/99 x 98 - 98/99 x 98 + 100/99 x 100 - 99/ 99 x 100
= 1 - 1/2 + 1/2 - 1/3 + 1/4 - 1/5 + ....... + 1/98 - 1/99 + 1/99 - 1/100
= 1 - 1/100
= 99/100.