Question

Tính:

1/1.2 + 1/2.3 + 1/3.4 +....+ 1/a.(a+1)

 

Giúp mình vs ạaa! 

Answer 1

\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{a\left(a+1\right)}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{a}-\dfrac{1}{a+1}\\ =\left(1-\dfrac{1}{a+1}\right)-\left(\dfrac{1}{2}-\dfrac{1}{2}\right)-\left(\dfrac{1}{3}-\dfrac{1}{3}\right)-...-\left(\dfrac{1}{a}-\dfrac{1}{a}\right)\\ =\left(\dfrac{a+1}{a+1}-\dfrac{1}{a+1}\right)-0-0-...-0\\ =\dfrac{a+1-1}{a+1}\\ =\dfrac{a}{a+1}\)