Question

CMR: Nếu \(\dfrac{a}{b}=\dfrac{c}{d}\) thì \(\left(\dfrac{a-b}{c-d}\right)^{2014}=\dfrac{a^{2014}+b^{2014}}{c^{2014}+d^{2014}}\)

Answer 1

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)\(\Rightarrow a=bk;c=dk\)

Xét \(VT=\left(\dfrac{a-b}{c-d}\right)^{2014}=\left(\dfrac{bk-b}{dk-d}\right)^{2014}=\left(\dfrac{b\left(k-1\right)}{d\left(k-1\right)}\right)^{2014}=\left(\dfrac{b}{d}\right)^{2014}\left(1\right)\)

Xét \(VP=\dfrac{a^{2014}+b^{2014}}{c^{2014}+d^{2014}}=\dfrac{b^{2014}k^{2014}+b^{2014}}{d^{2014}k^{2014}+d^{2014}}=\dfrac{b^{2014}\left(k^{2014}+1\right)}{d^{2014}\left(k^{2014}+1\right)}=\dfrac{b^{2014}}{d^{2014}}=\left(\dfrac{b}{d}\right)^{2014}\left(2\right)\)

Từ \(\left(1\right);\left(2\right)\) ta có ĐPCM