Chứng minh:
Đặt \(\dfrac{a}{2013}=\dfrac{a}{2014}=\dfrac{a}{2015}=k\)
\(\Rightarrow a=2013k,b=2014k,c=2015k\)
Vế trái
\(4\left(2013k-2014k\right).\left(2015k-2016k\right)\)\(=4.-k.-k=4k^2\)
Vế phải
\(\left(2015k-2013k\right)^2\)\(=\left(2k\right)^2=4k^2\)
\(\Rightarrow\)4(a−b).(b−c)=(c−a)\(\Rightarrow\)đpcm
Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:
\(\dfrac{a}{2013}=\dfrac{b}{2014}=\dfrac{c}{2015}=\dfrac{a-b}{2013-2014}=\dfrac{b-c}{2014-2015}=\dfrac{c-a}{2015-2013}\)\(\Rightarrow\dfrac{a-b}{-1}=\dfrac{b-c}{-1}=\dfrac{c-a}{2}\)
\(\Rightarrow\dfrac{a-b}{-1}.\dfrac{b-c}{-1}=\left(\dfrac{c-a}{2}\right)^2\)
\(\Rightarrow\dfrac{\left(a-b\right)\left(b-c\right)}{1}=\dfrac{\left(c-a\right)^2}{4}\)
\(\Rightarrow4\left(a-b\right)\left(b-c\right)=\left(c-a\right)^2\) (đpcm)
Chứng minh:
Đặt a2013=a2014=a2015=ka2013=a2014=a2015=k
⇒a=2013k,b=2014k,c=2015k⇒a=2013k,b=2014k,c=2015k
Vế trái
4(2013k−2014k).(2015k−2016k)4(2013k−2014k).(2015k−2016k)=4.−k.−k=4k2=4.−k.−k=4k2
Vế phải
(2015k−2013k)2(2015k−2013k)2=(2k)2=4k2=(2k)2=4k2
⇒⇒4(a−b).(b−c)=(c−a).(c-a) đpcm
