A)23/42-10/21

B)16/25-3/15

C)7/8-1/3-1/2

D)15/7-4/9-10/9

Vì \(b^2=ac\) ta suy ra \(\dfrac{a}{b}=\dfrac{b}{c}\). Đặt \(a=kb\) và \(b=kc\).

Khi đó \(\dfrac{a}{c}=\dfrac{k\left(kc\right)}{c}=k^2\). (1)

Từ tính chất của dãy tỉ số bằng nhau ta có:

\(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{2012b}{2012c}=\dfrac{a+2012b}{b+2012c}=k\), suy ra \(k^2=\dfrac{\left(a+2012b\right)^2}{\left(b+2012c\right)^2}\). (2)

Từ (1) và (2) suy ra \(k^2=\dfrac{a}{c}=\dfrac{\left(a+2012b\right)^2}{\left(b+2012c\right)^2}\) (đpcm)

b² = ac \(\Rightarrow\frac{a}{b}=\frac{b}{c}\)

Theo dãy tỉ số bằng nhau ta có :

\(\frac{a}{b}=\frac{b}{c}\Leftrightarrow\frac{a}{b}=\frac{2012b}{2012c}=\frac{a+2012b}{b+2012c}=\frac{\left(a+2012b\right)^2}{\left(b+2012c\right)^2}\)

....

\(b^2=ac\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{2012b}{2012c}=\frac{a+2012b}{b+2012c}\)

ta lại có:\(\frac{a}{b}\cdot\frac{b}{c}=\frac{\left(a+2012b\right)^2}{\left(b+2012c\right)^2}\)

\(=\frac{a}{c}=\frac{\left(a+2012b\right)^2}{\left(b+2012c\right)^2}\)(đpcm)

ĐỀ SAI NHA BẠN

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

=>\(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=\frac{d}{a}\)

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

=>\(\begin{cases}d=ak\\ c=dk=ak\cdot k=ak^2\\ b=ck=ak^2\cdot k=ak^3\\ a=bk=ak^3\cdot k=ak^4\end{cases}\Rightarrow ak^4-a=0\)

=>\(a\left(k^4-1\right)=0\)

=>\(k^4-1=0\)

=>\(k^4=1\)

=>k=1

=>a=b=c=d

\(A=\frac{2013a-2012b}{c+d}+\frac{2013b-2012c}{a+d}+\frac{2013c-2012d}{a+b}+\frac{2013d-2012a}{b+c}\)

\(=\frac{2013a-2012a}{a+a}+\frac{2013a-2012a}{a+a}+\frac{2013a-2012a}{a+a}+\frac{2013a-2012a}{a+a}\)

\(=\frac12+\frac12+\frac12+\frac12=\frac42=2\)