Question

Cho 3 số x;y;z thỏa mãn: \(\frac{x}{2012}=\frac{y}{2013}=\frac{z}{2014}\)

Chứng minh rằng \(\left(x-z\right)^3=8\left(x-y\right)^2\left(y-z\right)\)

Answer 1

Đặt \(\frac{x}{2012}=\frac{y}{2013}=\frac{z}{2014}=k\)=> \(\hept{\begin{cases}x=2012k\\y=2013k\\z=2014k\end{cases}}\)

khi đó, ta có: (x - z)³ =  (2012k - 2014k)³ = (-2k)³ = -8k³

 8(x - y)²(y - z) = 8(2012k - 2013k)²(2013 - 2014k) = 8(-k)².(-k) = -8k³

=> (x - z)³ = 8(x - y)²(y - z)