Question

CMR nếu \(\dfrac{a+2014}{a-2014}=\dfrac{b+2015}{b-2015}\) thì \(\dfrac{a}{2014}=\dfrac{b}{2015}\)

vs \(a\ne\pm2014;b\ne\pm2015\)

Answer 1

\(\dfrac{a+2014}{a-2014}=\dfrac{b+2015}{b-2015}=\dfrac{a+2014}{b+2015}=\dfrac{a-2014}{b-2015}\) (1)

Từ (1) \(\Rightarrow\dfrac{a+2014}{b+2015}=\dfrac{a-2014}{b-2015}=\dfrac{a+2014+a-2014}{b+2015+b-2015}\)

\(=\dfrac{a+2014-\left(a-2014\right)}{b+2015-\left(b-2015\right)}=\dfrac{2a}{2b}=\dfrac{4028}{4030}=\dfrac{a}{b}=\dfrac{2014}{2015}\) (2)

Từ (2) : \(\dfrac{a}{b}=\dfrac{2014}{2015}\Rightarrow\dfrac{a}{2014}=\dfrac{b}{2015}\) ( đpcm )

Answer 2

Ta có: \(\dfrac{a+2014}{a-2014}=\dfrac{b+2015}{b-2015}\) ( \(a\ne\pm2014;b\ne\pm2015\))

\(\Rightarrow\dfrac{a+2014}{b+2015}=\dfrac{a-2014}{b-2015}\)

Áp dụng tính chất của dãy tỉ số bằng nhau ta có:

\(\dfrac{a+2014}{b+2015}=\dfrac{a-2014}{b-2015}=\dfrac{\left(a+2014\right)+\left(a-2014\right)}{\left(b+2015\right)+\left(b-2015\right)}=\dfrac{\left(a+2014\right)-\left(a-2014\right)}{\left(b+2015\right)-\left(b-2015\right)}=\dfrac{a+2014+a-2014}{b+2015+b-2015}=\dfrac{a+2014-a+2014}{b+2015-b+2015}=\dfrac{2a}{2b}=\dfrac{4018}{2030}\)

\(\Rightarrow\dfrac{a}{b}=\dfrac{2014}{2015}\)

\(\Rightarrow\dfrac{a}{2014}=\dfrac{b}{2015}\) (đpcm)