Решите уравнение (x-1)/x+(x-2)/x+(x-3)/x+...+1/x=3, где х- целое число МЕТОДОМ АРИФМЕТИЧЕСКОЙ ПРОГРЕССИИ
Ответы
Ответ дал:
0
ОДЗ: х≠0
![a_{1}= frac{x-1}{x} a_{1}= frac{x-1}{x}](https://tex.z-dn.net/?f=a_%7B1%7D%3D+frac%7Bx-1%7D%7Bx%7D+)
![a_{2}= frac{x-2}{x} a_{2}= frac{x-2}{x}](https://tex.z-dn.net/?f=a_%7B2%7D%3D+frac%7Bx-2%7D%7Bx%7D+)
![d=a_{2}-a_{1}= frac{x-2}{x}- frac{x-1}{x}= frac{x-2-x+1}{x}=- frac{1}{x} d=a_{2}-a_{1}= frac{x-2}{x}- frac{x-1}{x}= frac{x-2-x+1}{x}=- frac{1}{x}](https://tex.z-dn.net/?f=d%3Da_%7B2%7D-a_%7B1%7D%3D+frac%7Bx-2%7D%7Bx%7D-+frac%7Bx-1%7D%7Bx%7D%3D+frac%7Bx-2-x%2B1%7D%7Bx%7D%3D-+frac%7B1%7D%7Bx%7D++++)
![a_{n}= frac{1}{x} a_{n}= frac{1}{x}](https://tex.z-dn.net/?f=a_%7Bn%7D%3D+frac%7B1%7D%7Bx%7D+)
![a_{n}=a_{1}+d(n-1) a_{n}=a_{1}+d(n-1)](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7D%2Bd%28n-1%29)
![frac{1}{x}= frac{x-1}{x}- frac{1}{x}(n-1) frac{1}{x}= frac{x-1}{x}- frac{1}{x}(n-1)](https://tex.z-dn.net/?f=+frac%7B1%7D%7Bx%7D%3D+frac%7Bx-1%7D%7Bx%7D-+frac%7B1%7D%7Bx%7D%28n-1%29+++)
![frac{1}{x} + frac{1}{x}(n-1)= frac{x-1}{x} frac{1}{x} + frac{1}{x}(n-1)= frac{x-1}{x}](https://tex.z-dn.net/?f=+frac%7B1%7D%7Bx%7D+%2B+frac%7B1%7D%7Bx%7D%28n-1%29%3D+frac%7Bx-1%7D%7Bx%7D++)
![frac{1}{x}(1+n-1)= frac{x-1}{x} frac{1}{x}(1+n-1)= frac{x-1}{x}](https://tex.z-dn.net/?f=+frac%7B1%7D%7Bx%7D%281%2Bn-1%29%3D+frac%7Bx-1%7D%7Bx%7D++)
![frac{n}{x}= frac{x-1}{x} frac{n}{x}= frac{x-1}{x}](https://tex.z-dn.net/?f=+frac%7Bn%7D%7Bx%7D%3D+frac%7Bx-1%7D%7Bx%7D++)
![n=x-1 n=x-1](https://tex.z-dn.net/?f=n%3Dx-1)
Sn=3
![S_{n}= frac{(a_{1}+a_{n})n}{2} S_{n}= frac{(a_{1}+a_{n})n}{2}](https://tex.z-dn.net/?f=S_%7Bn%7D%3D+frac%7B%28a_%7B1%7D%2Ba_%7Bn%7D%29n%7D%7B2%7D+)
![3= frac{( frac{x-1}{x}+ frac{1}{x} )*(x-1)}{2} 3= frac{( frac{x-1}{x}+ frac{1}{x} )*(x-1)}{2}](https://tex.z-dn.net/?f=3%3D+frac%7B%28+frac%7Bx-1%7D%7Bx%7D%2B+frac%7B1%7D%7Bx%7D++%29%2A%28x-1%29%7D%7B2%7D+)
![3*2= frac{x-1+1}{x}*(x-1) 3*2= frac{x-1+1}{x}*(x-1)](https://tex.z-dn.net/?f=3%2A2%3D+frac%7Bx-1%2B1%7D%7Bx%7D%2A%28x-1%29+)
6=x-1
x=6+1
x=7
Ответ: 7.
Sn=3
6=x-1
x=6+1
x=7
Ответ: 7.
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