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Ta có :\(\frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+d}\)
\(\Rightarrow\left(\frac{a}{b}\right)^6=\left(\frac{c}{d}\right)^6=\left(\frac{a+c}{b+d}\right)^6\)
\(\Rightarrow\frac{a^6}{b^6}=\frac{c^6}{d^6}=\frac{\left(a+c\right)^6}{\left(b+d\right)^6}\) (1)
Ta lại có : \(\frac{a^6}{b^6}=\frac{c^6}{d^6}=\frac{3a^6}{3b^6}=\frac{3a^6+c^6}{3b^6+d^6}\) (2)
Từ (1) và (2) \(\Rightarrow\frac{3a^6+c^6}{3b^6+d^6}=\frac{\left(a+c\right)^6}{\left(b+d\right)^6}\) (đpcm)
Đặt \(\frac{a}{b}=\frac{c}{d}\)=k \(\Rightarrow a=bk;c=dk\)
Ta có: \(\frac{3a^6+c^6}{3b^6+d^6}=\frac{3\left(bk\right)^6+\left(dk\right)^6}{3b^6+d^6}=\frac{3b^6.k^6+d^6.k^6}{3b^6+d^6}=\frac{k^6\left(3b^6+d^6\right)}{3b^6+d^6}=k^6\)(1)
\(\frac{\left(a+c\right)^6}{\left(b+d\right)^6}=\frac{\left(bk+dk\right)^6}{\left(b+d\right)^6}=\frac{\left[k\left(b+d\right)\right]^6}{\left(b+d\right)^6}=\frac{k^6.\left(b+d\right)^6}{\left(b+d\right)^6}=k^6\)(2)
Từ (1) và (2), ta có: \(\frac{3a^6+c^6}{3b^6+d^6}=\frac{\left(a+c\right)^6}{\left(b+d\right)^6}\)
\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a^6}{b^6}=\frac{c^6}{d^6}=\frac{3a^6}{3b^6}\)
áp dụng t/c dãy tỉ số bằng nhau ta có:
\(\frac{a^6}{b^6}=\frac{c^6}{d^6}=\frac{3a^6}{3b^6}=\frac{3a^6+c^6}{3b^6+d^6}\left(1\right)\)
\(\frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+d}\)
\(\Rightarrow\frac{a^6}{b^6}=\frac{c^6}{d^6}=\frac{\left(a+c\right)^6}{\left(b+d\right)^6}\left(2\right)\)
từ (1) và (2) => đpcm
+ Ta có
\(k=\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=\frac{a+b+c}{b+c+d}=k\Rightarrow\left(\frac{a+b+c}{b+c+d}\right)^3=k^3\) (1)
+ Ta có
\(k=\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\Rightarrow\left(\frac{a}{b}\right)^3=\left(\frac{b}{c}\right)^3=\left(\frac{c}{d}\right)^3=k^3\)
=> \(k^3=\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=\frac{a^3+b^3+c^3}{b^3+c^3+d^3}\) (2)
=> \(\left(\frac{a+b+c}{b+c+d}\right)^3.\left(\frac{a^3+b^3+c^3}{b^3+c^3+d^3}\right)=k^3.k^3=k^6\)
Ta có :
\(\frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+c}\Rightarrow\frac{a^6}{b^6}=\frac{c^6}{d^6}=\frac{\left(a+c\right)^6}{\left(b+d\right)^6}\) (1)
\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a^6}{b^6}=\frac{c^6}{d^6}=\frac{3a^6}{3b^6}=\frac{3a^6+c^6}{3b^6+d^6}\)(2)
Từ (1) ; (2) \(\Rightarrow\frac{\left(a+c\right)^6}{\left(b+d\right)^6}=\frac{3a^6+c^6}{3b^6+d^6}\) (đpcm)