Question

Tìm x \(\in\) Z biết

\(\dfrac{1}{1\cdot2}\) + \(\dfrac{1}{2\cdot3}\) + \(\dfrac{1}{3\cdot4}\) + ..... + \(\dfrac{1}{x\cdot\left(x+1\right)}\) = \(\dfrac{44}{45}\)

M.n giúp mình nha chiều phải nộp rùi !

Answer 1

\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+.......+\dfrac{1}{x\left(x+1\right)}=\dfrac{44}{45}\)

=> \(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}+......+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{44}{45}\)

=> \(1-\dfrac{1}{x+1}=\dfrac{44}{45}\)

=> \(\dfrac{x+1}{x+1}-\dfrac{1}{x+1}=\dfrac{44}{45}\)

=> \(\dfrac{x+1-1}{x+1}=\dfrac{44}{45}\)

=> \(\dfrac{x}{x+1}=\dfrac{44}{45}\)

=> 44(x+1)=45x

=> 44x+44=45x

=> 44x-45x=-44

=> -1x=-44

=> x=44

vậy x=44