Question

Tìm một số tự nhiên x biết: 1/2.3 + 1/3.4 + 1/4.5 +....+ 1/x.(x+1)=9/20

(chấm là nhân)

Answer 1

Có: \(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{x\left(x+1\right)}=\dfrac{9}{20}\)

\(\Rightarrow\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{x-1}-\dfrac{1}{x}+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{9}{20}\)

\(\Rightarrow\dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{9}{20}\)

\(\Rightarrow\dfrac{1}{x+1}=\dfrac{1}{20}\)

\(\Rightarrow x+1=20\Leftrightarrow x=19\)