Question

[lê thị lan anh]

Cho \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}\)

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

Answer 1

[Lê Thiên Anh]

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

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

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

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

Answer 2

[Hoàng Thị Ngọc Anh]

Đặt \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\)

\(\Rightarrow a=bk;b=ck;c=dk\) (1)

Thay (1) vào đề bài:

\(VT=\left(\frac{bk+ck+dk}{ck+dk+d}\right)^3=\left[\frac{k\left(c+d\right)+bk}{k\left(c+d\right)+d}\right]^3=\left(\frac{bk}{d}\right)^3=\frac{bk}{d}\)

\(VP=\frac{bk}{d}\)

\(\Rightarrow VT=VP\)

hay \(\left(\frac{a+b+c}{b+c+d}\right)^3=\frac{a}{d}\rightarrowđpcm.\)