Question

[nguyencris]

Cho 4 so nguyen duong a,b,c,d ma trong do b la trung binh cong cua 2 so a va c.Chung minh rang 4 so a,b,c,d lap thanh mot ti le thuc neu \(\frac{2}{c}\)=\(\frac{1}{b}+\frac{1}{d}\)

Answer 1

[I - Vy Nguyễn]

Ta có : \(b=\frac{a+c}{2}\) \(\implies\) \(2b=a+c\)

         \(\frac{2}{c}=\frac{1}{b}+\frac{1}{d}\) 

\(\implies\)  \(\frac{1}{2}.\frac{2}{c}=\frac{1}{2}.\left(\frac{1}{b}+\frac{1}{d}\right)\)

\(\implies\)  \(\frac{1}{c}=\frac{1}{2}.\left(\frac{1}{b}+\frac{1}{d}\right)\)

\(\iff\)  \(\frac{1}{c}=\frac{b+d}{2db}\)

        \(2db=c.\left(b+d\right)\)

  \(\left(a+c\right)d=cd+cb\)

     \(ad+cd=cd+cb\)

                 \(ad=cb\)

                 \(\frac{a}{c}=\frac{b}{d}\) là một tỉ lệ thức \(\left(đpcm\right)\)