Cho a/2014=b/2015=c/2016. CMR: 4(a-b)(b-c)=(c-a)2
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Dat \(\frac{a}{2014}=\frac{b}{2015}=\frac{c}{2016}=k\)
---> a = 2014k, b=2015k , c=2016k
VE trai : 4. ( 2014k-2015k). (2015k-2016k)=4. (-1k).(-1k)=4k2
Ve phai: (2016k-2014k)2=(2k)2=4k2
---> ve trai = ve phai----> dpcm
Ta có
\(\frac{a}{2014}=\frac{b}{2015}=\frac{c}{2016}=\frac{a-b}{-1}=\frac{b-c}{-1}=\frac{c-a}{2}\)
Suy ra
\(\left(\frac{a-b}{-1}\right)\left(\frac{b-c}{-1}\right)=\left(\frac{c-a}{2}\right)^2\)
-->\(\left(a-b\right)\left(b-c\right)=\left(c-a\right)^2\cdot2^2\)
-->\(\left(a-b\right)\left(b-c\right)=\left(c-a\right)^2\cdot4\)
-->\(4\left(a-b\right)\left(b-c\right)=\left(c-a\right)^2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(\frac{a}{2014}=\frac{b}{2015}=\frac{c}{2016}=k\)
\(\Rightarrow a=2014k;b=2015k;c=2016k\)
\(\Rightarrow4.\left(a-b\right).\left(b-c\right)\)
\(=4.\left(2014k-2015k\right).\left(2015k-2016k\right)\)
\(=4.\left(-k\right).\left(-k\right)\)
\(=4k^2\)(1)
Ta có: \(\left(c-a\right)^2=\left(2016k-2014k\right)^2=\left(2k\right)^2=4k^2\)(2)
Từ (1) và (2)
\(\Rightarrow4.\left(a-b\right)\left(b-c\right)=\left(c-a\right)^2\)
đpcm
Tham khảo nhé~
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![](https://rs.olm.vn/images/avt/0.png?1311)
Câu hỏi của Đỗ Thanh Uyên - Toán lớp 7 - Học toán với OnlineMath
Bạn tham khảo
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt:
\(\dfrac{a}{2014}=\dfrac{b}{2015}=\dfrac{c}{2016}=t\Leftrightarrow\left\{{}\begin{matrix}a=2014t\\b=2015t\\c=2016t\end{matrix}\right.\)
\(4\left(a-b\right)\left(b-c\right)=4\left(2014t-2015t\right)\left(2015t-2016t\right)=4\left(-k\right)\left(-k\right)=4k^2\)
\(\left(c-a\right)^2=\left(2k\right)^2=4k^2\)
Ta có đpcm