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Ta có:
f(x)=\(\frac{x^2+2x+1-x^2}{x^2\left(x+1\right)^2}=\frac{\left(x+1\right)^2-x^2}{x^2\left(x+1\right)^2}=\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\)
\(\Rightarrow f\left(1\right)=1-\frac{1}{2^2};f\left(2\right)=\frac{1}{2^2}-\frac{1}{3^2};...;f\left(x\right)=\frac{1}{x^2}-\frac{1}{\left(x-1\right)^2}\)
=> \(S=1-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{3^2}-\frac{1}{4^2}+...+\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}=1-\frac{1}{\left(x+1\right)^2}\)
Theo bài ra ta có :
\(1-\frac{1}{\left(x+1\right)^2}=\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-19+x\)
<=> \(1-\frac{1}{\left(x+1\right)^2}=2y\left(x+1\right)-\frac{1}{\left(x+1\right)^2}-19+x\)
<=> 1=2y(x+1)-19+x
<=> (2y+1)(x+1)=21
x, y thuộc N => 2y+1, x+1 thuộc N
Ta có bảng
| x+1 | 3 | 1 | 7 | 21 |
| 2y+1 | 7 | 21 | 3 | 1 |
| x | 2 | 0 | 6 | 20 |
| y | 3 | 10 | 1 | 0 |
Vậy....
Cô Linh Chi:
phần bảng x không có giá trị bằng 0
Nếu x = 0 thì hàm số f (x) có giá trị bằng 0
Lời giải:
Xét hàm \(f(x)=\frac{2x+1}{x^2(x+1)^2}\)
\(f(x)=\frac{x+(x+1)}{x^2(x+1)^2}=\frac{1}{x(x+1)^2}+\frac{1}{x^2(x+1)}=\frac{1}{x+1}(\frac{1}{x}-\frac{1}{x+1})+\frac{1}{x}(\frac{1}{x}-\frac{1}{x+1})\)
\(=\frac{1}{x^2}-\frac{1}{(x+1)^2}\)
Do đó:
\(s=f(1)+f(2)+f(3)+...+f(x)=1-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+...+\frac{1}{x^2}-\frac{1}{(x+1)^2}\)
\(=1-\frac{1}{(x+1)^2}\)
Để \(s=\frac{2y(x+1)^3-1}{(x+1)^2}-19+x\)
\(\Leftrightarrow 1-\frac{1}{(x+1)^2}=2y(x+1)-\frac{1}{(x+1)^2}-19+x\)
\(\Leftrightarrow 1=2y(x+1)-19+x\)
\(\Leftrightarrow (2y+1)(x+1)=21\)
Vì $x,y$ nguyên dương nen $2y+1$ và $x+1$ cũng là các nguyên dương lớn hơn $1$. Do đó ta xét các TH sau:
\(\left\{\begin{matrix} 2y+1=3\\ x+1=7\end{matrix}\right.\Rightarrow \left\{\begin{matrix} y=1\\ x=6\end{matrix}\right.\)
\(\left\{\begin{matrix} 2y+1=7\\ x+1=3\end{matrix}\right.\Rightarrow \left\{\begin{matrix} y=3\\ x=2\end{matrix}\right.\)
Vậy............
Câu hỏi của Nguyễn Bá Huy h - Toán lớp 7 - Học toán với OnlineMath
Em tham khảo nhé!
\(f\left(x\right)=\frac{2x+1}{x^2\left(x+1\right)^2}=\frac{x^2+2x+1-x^2}{x^2\left(x+1\right)^2}=\frac{\left(x+1\right)^2-x^2}{x^2\left(x+1\right)^2}\)
\(=\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\)
\(\Rightarrow f\left(1\right)=\frac{1}{1^2}-\frac{1}{2^2}\)
\(f\left(2\right)=\frac{1}{2^2}-\frac{1}{3^2}\)
\(f\left(3\right)=\frac{1}{3^2}-\frac{1}{4^2}\)
...
\(f\left(x\right)=\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\)
Lúc đó: \(f\left(1\right)+f\left(2\right)+f\left(3\right)+...+f\left(x\right)=\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{3^2}\)
\(-\frac{1}{4^2}+...+\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}=1-\frac{1}{\left(x+1\right)^2}\)
Thay về đầu bài, ta được: \(1-\frac{1}{\left(x+1\right)^2}=\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-19+x\)
\(\Leftrightarrow1-\frac{1}{\left(x+1\right)^2}=2y\left(x+1\right)-\frac{1}{\left(x+1\right)^2}-19+x\)
\(\Leftrightarrow2y\left(x+1\right)+\left(x+1\right)=21\)
\(\Leftrightarrow\left(x+1\right)\left(2y+1\right)=21\)
\(\Rightarrow\hept{\begin{cases}x+1\\2y+1\end{cases}}\inƯ\left(21\right)=\left\{\pm1;\pm3;\pm7;\pm21\right\}\)
Lập bảng:
| \(x+1\) | \(1\) | \(3\) | \(7\) | \(21\) | \(-1\) | \(-3\) | \(-7\) | \(-21\) |
| \(2y+1\) | \(21\) | \(7\) | \(3\) | \(1\) | \(-21\) | \(-7\) | \(-3\) | \(-1\) |
| \(x\) | \(0\) | \(2\) | \(6\) | \(20\) | \(-2\) | \(-4\) | \(-8\) | \(-22\) |
| \(y\) | \(10\) | \(3\) | \(1\) | \(0\) | \(-11\) | \(-4\) | \(-2\) | \(-1\) |
Mà \(x\ne0\)nên \(\left(x,y\right)\in\left\{\left(2,3\right);\left(6,1\right);\left(20,0\right);\left(-2,-11\right);\left(-4,-4\right);\left(-8,-2\right)\right\}\)\(\left(-22,-1\right)\)
\(\text{1)}\)
\(\text{Thay }x=-2,\text{ ta có: }f\left(-2\right)-5f\left(-2\right)=\left(-2\right)^2\Rightarrow f\left(-2\right)=-1\)
\(\Rightarrow f\left(x\right)=x^2+5f\left(-2\right)=x^2-5\)
\(f\left(3\right)=3^2-5\)
\(\text{2)}\)
\(\text{Thay }x=1,\text{ ta có: }f\left(1\right)+f\left(1\right)+f\left(1\right)=6\Rightarrow f\left(1\right)=2\)
\(\text{Thay }x=-1,\text{ ta có: }f\left(-1\right)+f\left(-1\right)+2=6\Rightarrow f\left(-1\right)=2\)
\(\text{3)}\)
\(\text{Thay }x=2,\text{ ta có: }f\left(2\right)+3f\left(\frac{1}{2}\right)=2^2\text{ (1)}\)
\(\text{Thay }x=\frac{1}{2},\text{ ta có: }f\left(\frac{1}{2}\right)+3f\left(2\right)=\left(\frac{1}{2}\right)^2\text{ (2)}\)
\(\text{(1) - 3}\times\text{(2) }\Rightarrow f\left(2\right)+3f\left(\frac{1}{2}\right)-3f\left(\frac{1}{2}\right)-9f\left(2\right)=4-\frac{1}{4}\)
\(\Rightarrow-8f\left(2\right)=\frac{15}{4}\Rightarrow f\left(2\right)=-\frac{15}{32}\)
a) Chỉ là thay số nên bạn tự làm nhé.
b) \(y_1=1\), \(y_2=f\left(y_1\right)=f\left(1\right)=1-\left|1\right|=0\), \(y_3=f\left(y_2\right)=f\left(0\right)=1-\left|0\right|=1\), cứ tiếp tục như vậy.
Dễ dàng nhận thấy rằng với \(k\)lẻ thì \(y_k=1\), \(k\)chẵn thì \(y_k=0\)(1).
Khi đó ta có:
\(A=y_1+y_2+...+y_{2021}\)
\(A=1+0+1+...+1\)
\(A=\frac{2021-1}{2}+1=1011\)
\(f\left(x\right)=\frac{x^2+2x+1-x^2}{x^2\left(x+1\right)^2}=\frac{\left(x+1\right)^2-x^2}{x^2\left(x+1\right)^2}=\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\)
\(\Rightarrow f\left(1\right)+f\left(2\right)+....+f\left(x\right)=1-\frac{1}{2^2}+\frac{1}{2^2}-....-\frac{1}{\left(x+1\right)^2}\)
\(\Rightarrow\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-19+x=\frac{x\left(x+2\right)}{\left(x+1\right)^2}\)
\(\Leftrightarrow\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-19+x=\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-20+\left(x+1\right)=\frac{x\left(x+2\right)}{\left(x+1\right)^2}\)
Dat:\(x+1=a\Rightarrow\frac{\left(2y+1\right)a^3-20a^2-1}{a^2}=\frac{a^2-1}{a^2}\Leftrightarrow\left(2y+1\right)a^3-20a^2-1=a^2-1\)
\(\Leftrightarrow\left(2y+1\right)a^3-20a^2=a^2\Leftrightarrow\left(2ay+a\right)-20=1\left(coi:x=-1cophailanghiemko\right)\)
\(\Leftrightarrow2ay+a=21\Leftrightarrow a\left(2y+1\right)=21\Leftrightarrow\left(x+1\right)\left(2y+1\right)=21\)