\(Q=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n}-\frac{1}{n+1}\)

\(Q=1-\frac{1}{n+1}=\frac{n}{n+1}\)

gọi d là UCLN của n,(n+1) ta có:

\(\hept{\begin{cases}n⋮d\\n+1⋮d\end{cases}\Rightarrow n+1-n⋮d\Rightarrow d=1}\)

=> Q là p/s tối giãn mà n khác 0 => Q ko thuộc Z

Áp dụng bđt bunhiacopxki ta được \(\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\le\left(1+1+1\right)^2=9\)

\(\Leftrightarrow\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\le10\)

bu-nhi a đâu phả vậy đâu bn?

BĐT cần chứng minh tương đương với : \(\frac{\left(x+z\right)^2}{xz}\ge\frac{y\left(x+z\right)}{xz}+\frac{x+z}{y}\)

\(\Leftrightarrow\frac{x+z}{xz}\ge\frac{y}{xz}+\frac{1}{y}\Leftrightarrow y\left(x+z\right)\ge y^2+xz\)

\(\Leftrightarrow y^2-y\left(x+z\right)+xz\le0\Leftrightarrow\left(y-x\right)\left(y-z\right)\le0\) ( luôn đúng vì \(z\ge y\ge x>0\))

Vậy BĐT đã được chứng minh khi x = y = z

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)

\(\frac{yz}{xyz}+\frac{xz}{xyz}+\frac{xy}{xyz}=0\)

\(\frac{yz+xz+xy}{xyz}=0\)

yz + xz + xy = 0

\(\left(x+y+z\right)^2=x^2+y^2+z^2+2xy+2xz+2yz=x^2+y^2+z^2+2\times\left(xy+xz+yz\right)=x^2+y^2+z^2+2\times0=x^2+y^2+z^2\left(\text{đ}pcm\right)\)

a) Từ giả thiết suy ra: xy + yz + zx = 0

Do đó:

\(\left(x+y+z\right)^2=x^2+y^2+z^2+2\left(xy+yz+zx\right)=x^2+y^2+z^2\)

b) Đặt \(\frac{1}{a-b}=x\); \(\frac{1}{b-c}=y\); \(\frac{1}{c-a}=z\)

Ta có: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=a-b+b-c+c-a=0\)

Theo câu a ta có: \(x^2+y^2+z^2=\left(x+y+z\right)^2\)

Suy ra điều phải chứng minh

a)

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)

\(\Rightarrow\frac{xy+yz+xz}{xyz}=0\)

\(\Rightarrow xy+yz+xz=0\)

\(x^2+y^2+z^2=\left(x+y+z\right)^2\)

\(\Rightarrow x^2+y^2+z^2=x^2+y^2+z^2+2xy+2yz+2xz\)

\(\Rightarrow x^2+y^2+z^2=x^2+y^2+z^2+2\left(xy+yz+xz\right)\)

Do \(xy+yz+xz=0\)

\(\Rightarrow x^2+y^2+z^2=x^2+y^2+z^2\) ( đpcm )

b)

\(\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}=\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)^2\)

\(\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}=\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}+\frac{2}{\left(a-b\right)\left(b-c\right)}+\frac{2}{\left(b-c\right)\left(c-a\right)}+\frac{2}{\left(a-b\right)\left(c-a\right)}\)

\(\Rightarrow\frac{2}{\left(a-b\right)\left(b-c\right)}+\frac{2}{\left(b-c\right)\left(c-a\right)}+\frac{2}{\left(a-b\right)\left(c-a\right)}=0\)

\(\Rightarrow2\left(\frac{1}{\left(a-b\right)\left(b-c\right)}+\frac{1}{\left(b-c\right)\left(c-a\right)}+\frac{1}{\left(a-b\right)\left(c-a\right)}\right)=0\)

\(\Rightarrow\frac{1}{\left(a-b\right)\left(b-c\right)}+\frac{1}{\left(b-c\right)\left(c-a\right)}+\frac{1}{\left(a-b\right)\left(c-a\right)}=0\)

\(\Rightarrow\frac{\left(c-a\right)^2\left(b-c\right)\left(a-b\right)+\left(a-b\right)^2\left(b-c\right)\left(c-a\right)+\left(b-c\right)^2\left(a-b\right)\left(c-a\right)}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}=0\)

\(\Rightarrow\frac{\left(c-a\right)\left(b-c\right)\left(a-b\right)\left[\left(a-b\right)+\left(b-c\right)+\left(c-a\right)\right]}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}=0\)

\(\Rightarrow\frac{\left(c-a\right)\left(b-c\right)\left(a-b\right)\left[\left(-a+a\right)+\left(-b+b\right)+\left(-c+c\right)\right]}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}=0\)

\(\Rightarrow\frac{\left(c-a\right)\left(b-c\right)\left(a-b\right).0}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}=0\)

\(\Rightarrow0=0\) ( đpcm )

1. D= 1/3 + 1/3.4 + 1/3.4.5 + 1/3.4.5....n < 1/2 + 1/3.4 + 1/4.5 + ...+ 1/ n.(n-1)

=> còn lại thì bạn có thể tự chứng minh

mk chả hiểu j