Call A=\(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{999+1000}\)

We have A=\(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{999\cdot1000}\)

\(\Rightarrow\)A=\(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{999}-\dfrac{1}{1000}\)

\(\Rightarrow\)A=\(1-\left(\dfrac{1}{2}-\dfrac{1}{2}\right)-\left(\dfrac{1}{3}-\dfrac{1}{3}\right)-\left(\dfrac{1}{4}-\dfrac{1}{4}\right)-...-\left(\dfrac{1}{999}-\dfrac{1}{999}\right)-\dfrac{1}{1000}\)\(\Rightarrow\) A=\(1-\dfrac{1}{1000}=\dfrac{1000}{1000}-\dfrac{1}{1000}=\dfrac{999}{1000}\)

Call A=a+b-c

We have |a|=2 \(\Rightarrow\) a = 2 or a = (-2)

|b|= 3 \(\Rightarrow\) b = 3 or b = (-3)

|c|= 4 \(\Rightarrow\) c = 4 or c = (-4)

\(\Rightarrow\)a = 2; b = 3;c = 4 or a = -2;b = -3;c = -4

But a > b > c ( theme for )
So a = -2;b = -3;c = -4 ( because -2 > -3 > -4 )

Substitute a = -2;b = -3;c = -4 to A, we have :

A= -2+(-3)-(-4)

\(\Rightarrow\)A= -2-3+4

\(\Rightarrow\)A= -5+4

\(\Rightarrow\)A= -1 

\(\Rightarrow\) a+b-c= -1 in a = -2;b = -3;c = -4