Question

\(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}.CMR\left(\dfrac{a+b+c}{b+c+d}\right)^3=\dfrac{a}{d}\)

Answer 1

Từ \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}=\dfrac{a+b+c}{b+c+d}=\left(\dfrac{a+b+c}{b+c+d}\right)^3\)

Có \(\dfrac{a}{b}.\dfrac{a}{b}.\dfrac{a}{b}=\dfrac{a}{b}.\dfrac{b}{c}.\dfrac{c}{d}=\dfrac{a}{d}\left(2\right)\)

Từ (1) và (2) \(\Rightarrow\left(\dfrac{a+b+c}{b+c+d}\right)^3=\dfrac{a}{d}\left(dpcm\right)\)

Answer 2

Ta có: \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}\) \(\Rightarrow\left(\dfrac{a}{b}\right)^3=\left(\dfrac{b}{c}\right)^3=\left(\dfrac{c}{d}\right)^3=\dfrac{abc}{bcd}=\dfrac{a}{d}\)

\(\Rightarrow\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}=\dfrac{a}{d}\) (1)

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

\(\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}=\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}\) (2)

Từ (1) và (2) suy ra: \(\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}=\dfrac{a}{d}\)

Answer 3

Ta có: \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}=\dfrac{a+b+c}{b+c+d}\)

\(\Leftrightarrow\dfrac{a^3}{b^3}=\dfrac{\left(a+b+c\right)^3}{\left(b+c+d\right)^3}\)

\(\Leftrightarrow\dfrac{a}{b}.\dfrac{b}{c}.\dfrac{c}{d}=\dfrac{\left(a+b+c\right)^3}{\left(b+c+d\right)^3}\)

\(\Leftrightarrow\dfrac{a}{d}=\dfrac{\left(a+b+c\right)^3}{\left(b+c+d\right)^3}\left(đpcm\right)\)

Vậy .................

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