Question

cho a,b,c\(\in\)R và a,b,c\(\ne\)0 thỏa mãn b² = a.c. c/m rằng

\(\dfrac{a}{c}=\dfrac{(a+2012.b)^2}{(b+2012.c)^2}\)

Answer 1

\(b^2=a\cdot c\Rightarrow\dfrac{a}{b}=\dfrac{b}{c}\)

\(đặt\):\(\dfrac{a}{b}=\dfrac{b}{c}=k,ta\) \(có\):\(a=bk;b=ck\)

\(\dfrac{a}{c}=\dfrac{bk}{c}=\dfrac{ck+k}{c}=k^2\left(1\right)\)

\(\left(\dfrac{a+2012b}{b+2102c}\right)^2=\left(\dfrac{bk+2012b}{ck+2012c}\right)^2=\left(\dfrac{b\left(k+2012\right)}{c\left(k+2012\right)}\right)^2=\left(\dfrac{b}{c}\right)^2=k^2\left(2\right)\)Từ \(\left(1\right)và\left(2\right)\Rightarrow\dfrac{a}{c}=\left(\dfrac{a+2012b}{b+2012c}\right)^2\left(đpcm\right)\)