Question

Cho đẳng thức :\(x\times\left(x+1\right)\times\left(x+2\right)\times.............\times\left(x+2016\right)=2016\)(với x>0)

Chứng tỏ rằng \(x< \dfrac{1}{2015!}\)

Answer 1

Ta có \(x=\dfrac{2016}{x\times\left(x+1\right)\times\left(x+2\right)\times........\times\left(x+2016\right)}\)

\(\dfrac{1}{2015!}=\dfrac{2016}{2016!}=\dfrac{2016}{1\times2\times...........\times2016}\)

Vì x > 0=> \(\left(x+1\right)\times\left(x+2\right)\times...\times\left(x+2016\right)>1\times2\times...\times2016\)

\(\Rightarrow\dfrac{1}{\left(x+1\right)\times\left(x+2\right)\times.......\times\left(x+2016\right)}< \dfrac{1}{1\times2\times..........\times2016}\)\(\Rightarrow\dfrac{2016}{\left(x+1\right)\times\left(x+2\right)\times.......\times\left(x+2016\right)}< \dfrac{2016}{1\times2\times......\times2016}\)

\(\Leftrightarrow x< \dfrac{1}{2015!}\)(đpcm)

Answer 2

Ta có \(x=\dfrac{2016}{\left(x+1\right)\times\left(x+2\right)\times....\times\left(x+2016\right)}\)

\(\dfrac{1}{2015!}=\dfrac{2016}{2016!}=\dfrac{2016}{1\times2\times.....\times2016}\)

Vì x>0=>(x+1)×(x+2)×.............×(x+2016) >\(1\times2\times.....\times2016\)

\(\Rightarrow\dfrac{1}{\left(x+1\right)\times\left(x+2\right)\times......\times\left(x+2016\right)}>\dfrac{1}{1\times2\times......\times2016}\)

\(\Rightarrow\dfrac{2016}{\left(x+1\right)\times\left(x+2\right)\times......\times\left(x+2016\right)}>\dfrac{2016}{1\times2\times......\times2016}\)

\(\Leftrightarrow x< \dfrac{1}{2015!}\)(đpcm)