Question

[Lê Ngọc Duyên]

Tìm x, biết:\(\left(1+5+5^2+5^3+...+5^{2016}\right).\left|x-1\right|=5^{2017}-1\)

Answer 1

[Lấp La Lấp Lánh]

Đặt \(S=1+5+5^2+5^3+...+5^{2016}\)

\(\Rightarrow5S=5+5^2+5^3+...+5^{2017}\)

\(\Rightarrow4S=5S-S=5+5^2+...+5^{2017}-1-5-...-5^{2016}=5^{2017}-1\)

\(\Rightarrow S=\dfrac{5^{2017}-1}{4}\)

Theo đề bài ta được: \(S.\left|x-1\right|=5^{2017}-1\)

\(\Leftrightarrow\dfrac{5^{2017}-1}{4}.\left|x-1\right|=5^{2017}-1\Leftrightarrow\dfrac{\left|x-1\right|}{4}=1\)

\(\Leftrightarrow\left|x-1\right|=4\Leftrightarrow\left[{}\begin{matrix}x-1=4\\x-1=-4\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=5\\x=-3\end{matrix}\right.\)