ta có:

1-1/2+1/2-1/3+1/3-1/4+....+1/x -1/x+1 =499/500

1-1/x+1 =499/500

1/x+1 =1/500 

x+1=500

x=499

\(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{X\times\left(X+1\right)}=\frac{499}{500}\)

\(\Leftrightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{X}-\frac{1}{X+1}=\frac{499}{500}\)

\(\Leftrightarrow1-\frac{1}{X+1}=\frac{499}{500}\)

\(\Leftrightarrow\frac{1}{X+1}=\frac{1}{500}\)

\(\Leftrightarrow X+1=500\)

\(\Leftrightarrow X=499\)

\(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-....-\frac{1}{x+1}=\frac{499}{500}\)

1 - 499/500 = 1/x + 1

1/500 = 1/x+1

x + 1 = 500

x = 499

a) 1/1.2 + 1/2.3 + 1/3.4 + .... + 1/x.(x+1) = 499/500

1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + .... + 1/x - 1/x+1 = 499/500

1 - 1/x+1 = 499/500

1/x+1 = 1 - 499/500

1/x+1 = 1/500

x + 1 = 500

     x = 500 - 1 

     x = 499

b) 1/1.3 + 1/3.5 + 1/5.7 + .... + 1/x.(x+2) = 20/41

1/2 . [ 2/1.3 + 2/3.5 + 2/5.7 + ... + 2/x.(x+2) ] = 20/41

1/2 . [ 1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + ... + 1/x - 1/x+2 ] = 20/41

1/2 . [ 1 - 1/x+2 ) = 20/41

1 - 1/x+2 = 20/41 : 1/2

1 - 1/x+2 = 40/41

1/x+2 = 1 - 40/41

1/x+2 = 1/41

x + 2 = 41

      x = 41 - 2 

      x = 39

\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+....+\frac{2}{x\left(x+1\right)}=\frac{499}{999}\)

\(\Leftrightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+....+\frac{2}{x\left(x+1\right)}=\frac{499}{999}\)

\(\Leftrightarrow\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+....+\frac{2}{x\left(x+1\right)}=\frac{499}{999}\)

\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{499}{999}\)

\(\Leftrightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{499}{999}\div2\)

\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{499}{1998}\)

\(\Leftrightarrow\frac{1}{x+1}=\frac{250}{999}\)

\(\Leftrightarrow\left(x+1\right).250=999\Rightarrow x+1=\frac{999}{250}\Rightarrow x=\frac{999}{250}-1=\frac{749}{250}\)

Như kiểu đề sai hay sao ý 

Bài 1

a) 3 2/5 - 1/2

= 17/5 - 1/2

= 34/10 - 5/10

= 29/10

b) 4/5 + 1/5 × 3/4

= 4/5 + 3/20

= 16/20 + 3/20

= 19/20

c) 3 1/2 × 1 1/7

= 7/2 × 8/7

= 4

d) 4 1/6 : 2 1/3

= 25/6 : 7/3

= 25/14

Bài 2

a) 3 × 1/2 + 1/4 × 1/3

= 3/2 + 1/12

= 18/12 + 1/12

= 19/12

b) 1 4/5 - 2/3 : 2 1/3

= 9/5 - 2/3 : 7/3

= 9/5 - 2/7

= 63/35 - 10/35

= 53/35

1 / 1 x 2 = 1-1/2 và 1/2x3 = (1/2)-(1/3) tương tự đến 1/(x-1).x=(1/x-1)-(1/x). Cuối cùng ta có phép tính 1+(1/x-1)-(1/x)=15/16

1 / 1 x 2 = 1-1/2 và 1/2x3 = (1/2)-(1/3) tương tự đến 1/(x-1).x=(1/x-1)-(1/x).

Cuối cùng ta có phép tính

1+(1/x-1)-(1/x)=15/16.