Tìm x biết
1/3+1/6+1/10+...+2/x(x+1)=2013/2014
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1/3 = 2/6 = 2/(2x3) = 2/2 - 2/3 1/6 = 2/12 = 2/(3x4) = 2/3 - 2/4 ... 2/x(x + 1) = 2/x - 2/(x +1) Do đó: 1/3 + 1/6 + ... + 2/x(x+1) = 2/2 - 2/3 + 2/3 - 2/4 + ... +2/x - 2/(x + 1) = 2/2 - 2/(x+1) suy ra 1 - 2/(x + 1) = 2013/2014 x= 4027
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài 3.
\(\left\{{}\begin{matrix}a\left(a+b+c\right)=-\dfrac{1}{24}\left(1\right)\\c\left(a+b+c\right)=-\dfrac{1}{72}\left(2\right)\\b\left(a+b+c\right)=\dfrac{1}{16}\left(3\right)\end{matrix}\right.\)
Dễ thấy \(a,b,c\ne0\Rightarrow a+b+c\ne0\)
Chia (1) cho (2), ta được \(\dfrac{a}{c}=3\Rightarrow a=3c\left(4\right)\)
Chia (2) cho (3) ta được: \(\dfrac{c}{b}=-\dfrac{2}{9}\Rightarrow b=-\dfrac{9}{2}c\left(5\right)\).
Thay (4), (5) vào (2), ta được: \(-\dfrac{1}{2}c^2=-\dfrac{1}{72}\)
\(\Rightarrow c=\pm\dfrac{1}{6}\).
Với \(c=\dfrac{1}{6}\Rightarrow\left\{{}\begin{matrix}a=3c=\dfrac{1}{2}\\b=-\dfrac{9}{2}c=-\dfrac{3}{4}\end{matrix}\right.\)
Với \(c=-\dfrac{1}{6}\Rightarrow\left\{{}\begin{matrix}a=3c=-\dfrac{1}{2}\\b=-\dfrac{9}{2}c=\dfrac{3}{4}\end{matrix}\right.\)
Vậy: \(\left(a;b;c\right)=\left\{\left(\dfrac{1}{2};-\dfrac{3}{4};\dfrac{1}{6}\right);\left(-\dfrac{1}{2};\dfrac{3}{4};-\dfrac{1}{6}\right)\right\}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
1. x(x + 1) - x2 + 1 = 0
<=> x(x + 1) - (x2 - 1) = 0
<=> x(x + 1) - (x + 1)(x - 1) = 0
<=> (x - x + 1)(x + 1) = 0
<=> x + 1 = 0\
<=> x = -1
2. 4x(x - 2) - 6 + 3x = 0
<=> 4x(x - 2) - (3x - 6) = 0
<=> 4x(x - 2) - 3(x - 2) = 0
<=> (4x - 3)(x - 2) = 0
<=> \(\left[{}\begin{matrix}4x-3=0\\x-2=0\end{matrix}\right.\)
<=> \(\left[{}\begin{matrix}x=\dfrac{3}{4}\\x=2\end{matrix}\right.\)
3. x(x + 2) - 3(x + 2) = 0
<=> (x - 3)(x + 2) = 0
<=> \(\left[{}\begin{matrix}x-3=0\\x+2=0\end{matrix}\right.\)
<=> \(\left[{}\begin{matrix}x=3\\x=-2\end{matrix}\right.\)
![](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)
![](https://rs.olm.vn/images/avt/0.png?1311)
1/3 = 2/6 = 2/(2x3) = 2/2 - 2/3
1/6 = 2/12 = 2/(3x4) = 2/3 - 2/4
...
2/x(x + 1) = 2/x - 2/(x +1)
Do đó:
1/3 + 1/6 + ... + 2/x(x+1) = 2/2 - 2/3 + 2/3 - 2/4 + ... +2/x - 2/(x + 1) = 2/2 - 2/(x+1)
suy ra 1 - 2/(x + 1) = 2013/2014
x= 4027