Question

Cho \(\dfrac{a+c}{b+d}=\dfrac{a-c}{b-d}\) (Với a, b,c,d khác 0 và b khác d ,-d)

CMR \(\dfrac{a^{2009}-c^{2009}}{b^{2009}-d^{2009}}=\left(\dfrac{a}{b}\right)^{2009}\)

Answer 1

\(\dfrac{a+c}{b+d}=\dfrac{a-c}{b-d}\)

\(\Leftrightarrow ab-ad+cb-cd=ab+ad-cb-cd\)

=>-2ad=-2cb

=>ad=cb

=>a/b=c/d

Đặt a/b=c/d=k

=>a=bk; c=dk

\(\dfrac{a^{2009}-c^{2009}}{b^{2009}-d^{2009}}=\dfrac{b^{2009}k^{2009}-d^{2009}k^{2009}}{b^{2009}-d^{2009}}=k^{2009}\)

\(\left(\dfrac{a}{b}\right)^{2009}=\left(\dfrac{bk}{b}\right)^{2009}=k^{2009}\)

Do đó: \(\dfrac{a^{2009}-c^{2009}}{b^{2009}-d^{2009}}=\left(\dfrac{a}{b}\right)^{2009}\)