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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 \(\left(x,y\right)\in\left\{\left(2,3\right);\left(6,1\right);\left(20,0\right);\left(-2,-11\right);\left(-4,-4\right);\left(-8,-2\right)\right\}\)\(\left(-22,-1\right)\)
Giả sử f(x)=ax2+bx+cf(x)=ax2+bx+c (do đề bài cho là đa thức bậc hai)
Suy ra
f(x)−f(x−1)=ax2+bx+c−a(x−1)2−b(x−1)−c=2ax+a+bf(x)−f(x−1)=ax2+bx+c−a(x−1)2−b(x−1)−c=2ax+a+b
Mà f(x)−f(x−1)=xf(x)−f(x−1)=x
⇒2ax+a+b=x⇒2ax+a+b=x
Do đó a+b=0a+b=0 và a=1/2a=1/2 từ đó ta suy ra a=1/2;b=−1/2a=1/2;b=−1/2
Do đó f(x)=x22−x2+cf(x)=x22−x2+c
f(n)=1+2+3+...+nf(n)=1+2+3+...+n
Áp dụng điều ta vừa chứng minh được thì:
f(1)−f(0)=1f(1)−f(0)=1
f(2)−f(1)=2f(2)−f(1)=2
....
f(n)−f(n−1)=nf(n)−f(n−1)=n
Do đó
1+2+...+n=f(1)−f(0)+f(2)−f(1)+...+f(n)−f(n−1)=f(n)−f(0)=n22−n2=n(n−1)2
Suy ra
f(x)−f(x−1)=ax2+bx+c−a(x−1)2−b(x−1)−c=2ax+a+bf(x)−f(x−1)=ax2+bx+c−a(x−1)2−b(x−1)−c=2ax+a+b
Mà f(x)−f(x−1)=xf(x)−f(x−1)=x
⇒2ax+a+b=x⇒2ax+a+b=x
Do đó a+b=0a+b=0 và a=1/2a=1/2 từ đó ta suy ra a=1/2;b=−1/2a=1/2;b=−1/2
Do đó f(x)=x22−x2+cf(x)=x22−x2+c
f(n)=1+2+3+...+nf(n)=1+2+3+...+n
Áp dụng điều ta vừa chứng minh được thì:
f(1)−f(0)=1f(1)−f(0)=1
f(2)−f(1)=2f(2)−f(1)=2
....
f(n)−f(n−1)=nf(n)−f(n−1)=n
Do đó
1+2+...+n=f(1)−f(0)+f(2)−f(1)+...+f(n)−f(n−1)=f(n)−f(0)=n22−n2=n(n−1)2
:3
\(f\left(x\right)=\frac{x^2+2x+1-x^2}{x^2\left(x+1\right)^2}=\frac{\left(x+1\right)^2-x^2}{x^2\left(x+1\right)^2}=\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\)
\(\Rightarrow f\left(1\right)+f\left(2\right)+....+f\left(x\right)=1-\frac{1}{2^2}+\frac{1}{2^2}-....-\frac{1}{\left(x+1\right)^2}\)
\(\Rightarrow\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-19+x=\frac{x\left(x+2\right)}{\left(x+1\right)^2}\)
\(\Leftrightarrow\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-19+x=\frac{2y\left(x+1\right)^3-1}{\left(x+1\right)^2}-20+\left(x+1\right)=\frac{x\left(x+2\right)}{\left(x+1\right)^2}\)
Dat:\(x+1=a\Rightarrow\frac{\left(2y+1\right)a^3-20a^2-1}{a^2}=\frac{a^2-1}{a^2}\Leftrightarrow\left(2y+1\right)a^3-20a^2-1=a^2-1\)
\(\Leftrightarrow\left(2y+1\right)a^3-20a^2=a^2\Leftrightarrow\left(2ay+a\right)-20=1\left(coi:x=-1cophailanghiemko\right)\)
\(\Leftrightarrow2ay+a=21\Leftrightarrow a\left(2y+1\right)=21\Leftrightarrow\left(x+1\right)\left(2y+1\right)=21\)
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
Xét \(f(x^2)=0\) => \(3x-1/2=0 =>3x=1/2 =>x=\)\(\frac{1}{2}\cdot\frac{1}{3}=\frac{1}{6}\)
Ta có: \(f\left(x\right)=ax^2+bx+c\)
\(\Rightarrow\left\{{}\begin{matrix}f\left(2\right)=a\cdot2^2+2b+c=4a+2b+c\\f\left(-5\right)=a\cdot\left(-5\right)^2-5b+c=25a-5b+c\end{matrix}\right.\)
\(\Rightarrow f\left(2\right)\cdot f\left(-5\right)=\left(4a+2b+c\right)\left(25a-5b+c\right)\)
Lại có:\(25a-5b+c=29a+2c-c-4a-5b\)
\(=3b-c-4a-5b=-2b-c-4a=-\left(4a+2b+c\right)\)
\(\Rightarrow f\left(2\right)\cdot f\left(-5\right)=-\left(4a+2b+c\right)\left(4a+2b+c\right)\)
\(=-\left(4a+2b+c\right)^2\le0\forall a,b,c\)