Question

Tính nhanh : \(\dfrac{1}{1}.\dfrac{1}{2}+\dfrac{1}{2}.\dfrac{1}{3}+\dfrac{1}{3}.\dfrac{1}{4}+...+\dfrac{1}{998}.\dfrac{1}{999}+\dfrac{1}{999}.\dfrac{1}{1000}\)

Answer 1

\(\dfrac{1}{1}.\dfrac{1}{2}+\dfrac{1}{2}.\dfrac{1}{3}+\dfrac{1}{3}.\dfrac{1}{4}+...+\dfrac{1}{999}.\dfrac{1}{1000}\\ =\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{999.1000}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{999}-\dfrac{1}{1000}\\ =1-\dfrac{1}{1000}=\dfrac{999}{1000}\)

Answer 2

ta có

1/1.1/2=1-1/2

1/2.1/3=1/2-1/3

1/3.1/4=1/3-1/4

............

1/999.1/1000=1/999-1/1000

Từ đó suy ra

1/1.1/2+1/2-1/3+1/3+.......+1/998.1/999+1/999.1/1000

=1/1-1/2+1/2-1/3+1/3-.....+1/998-1/999+1/999-1/1000

=1-1/1000

=1000/1000-1/1000

=999/1000

nhớ like bạn nhé