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

A nguyên nên \(3⋮n-2\). Vậy \(n-2\in\left(1,-1,3,-3\right)\Rightarrow n\in\left(3,1,5,-1\right)\)thì A nguyên.

2. a,Ta cần CM  \(\frac{a}{b}< \frac{a+c}{b+c}\Rightarrow a\left(b+c\right)< b\left(a+c\right)\Rightarrow ab+ac< ab+bc\Rightarrow ac< bc\)(luôn đúng)

Suy ra điều phải chứng minh.

b, Có: \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}>\frac{a}{a+b+c}+\frac{b}{a+b+c}+\frac{c}{a+b+c}=\frac{a+b+c}{a+b+c}=1\)

Có:(suy ra từ phần a) \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< \frac{a+c}{a+b+c}+\frac{b+a}{a+b+c}+\frac{c+b}{a+b+c}=\frac{2\left(a+b+c\right)}{a+b+c}=2\)

Vậy \(1< \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2\)

BẤM ĐÚNG CHO MÌNH, KO THÌ LẦN SAU KO GIÚP NỮA

Để \(A=\frac{n+1}{n-2}\)có giá trị nguyên => n + 1 chia hết cho n-2

\(=>\left(n-2\right)+3⋮\)\(n-2\)

Mà \(\left(n-2\right)⋮\)\(n-2\)

\(=>3⋮\)\(n-2\)

\(=>n-2\inƯ\left(3\right)=\){1;-1;3;-3}

Ta có bảng :

Vậy \(n\in\){3;1;5;-1} để \(A=\frac{n+1}{n-2}\in Z\)

Câu 2 :

a)Vì \(a< b=>\frac{a}{b}< 1\)

Ta so sánh : \(\frac{a}{b}\)và \(\frac{a+c}{b+c}\)

Ta có : \(\frac{a}{b}=\frac{a.\left(b+c\right)}{b.\left(b+c\right)}=\frac{ab+ac}{b.\left(b+c\right)}\)

             \(\frac{a+c}{b+c}=\frac{b.\left(a+c\right)}{b.\left(b+c\right)}=\frac{ab+bc}{b.\left(b+c\right)}\)

Vì a,b,c \(\in\)N* mà a<b => \(ab+ac< ab+bc=>\frac{ab+ac}{b.\left(b+c\right)}< \frac{ab+bc}{b.\left(b+c\right)}=>\frac{a}{b}< \frac{a+c}{b+c}\left(đpcm\right)\)

b) Ta có : \(\frac{a}{a+b}>\frac{a}{a+b+c}\)

                \(\frac{b}{b+c}>\frac{b}{a+b+c}\)

                  \(\frac{c}{a+c}>\frac{c}{a+b+c}\)

\(=>\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}>\frac{a}{a+b+c}+\frac{b}{a+b+c}+\frac{c}{a+b+c}=\frac{a+b+c}{a+b+c}=1\)

\(=>\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}>1\)                                                    \(\left(1\right)\)

Vì a,b,c \(\in\)N* mà a<b => \(\frac{a}{a+b}< 1\)=>\(\frac{a}{a+b}< \frac{a+c}{a+b+c}\)

Tương tự như vậy ta có \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}< \frac{a+c}{a+b+c}+\frac{a+b}{a+b+c}+\frac{c+b}{a+b+c}=\frac{a+c+b+a+c+b}{a+b+c}=\frac{2a+2b+2c}{a+b+c}=2\)

\(=>\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}< 2\)                                                                    \(\left(2\right)\)

 Từ \(\left(1\right);\left(2\right)=>1< \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{a+c}< 2\left(đpcm\right)\)

Ta có :

\(\frac{30}{43}=\frac{1}{\frac{43}{30}}=\frac{1}{1+\frac{13}{30}}=\frac{1}{1+\frac{1}{\frac{30}{13}}}=\frac{1}{1+\frac{1}{2+\frac{4}{13}}}=\frac{1}{1+\frac{1}{2+\frac{1}{\frac{13}{4}}}}=\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}}\)

Vậy \(a=1;b=2;c=3;d=4\)

Ta có: \(\frac{30}{43}=\frac{1}{\frac{43}{30}}=\frac{1}{1+\frac{13}{30}}=\frac{1}{1+\frac{1}{2+\frac{4}{13}}}=\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}}\)

\(\Rightarrow\)a = 1 ; b  = 2 ; c = 3 ; d = 4

Vậy: 

a = 1 ; b  = 2 ; c = 3 ; d = 4

Sao bạn kia đăng lên rồi tự giải vậy ??????????????????????????????

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

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

mà \(\left(\frac{1}{ab}+\frac{1}{\left(a+b+c\right)c}\right)\ne0\)với mọi a,b,c

\(\Rightarrow\)a+b=0\(\Leftrightarrow\)a=-b là hai số đối nhau (1)

từ đó được \(a^n=-b^n\)với mọi n lẻ.

Khi đó \(\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}\Leftrightarrow\frac{1}{c^n}=\frac{1}{c^n}\)luôn đúng (2)

Từ (1)và(2) ta được đpcm

sao bn toàn cây khó thế?

a) 3x - 6y = 3 . x - 3 . 2y = 3(x - 2y)

b) 2525x² + 5x³ + x²y = x² (2525 + 5x + y)

c) 14x²y – 21xy² + 28x²y² = 7xy . 2x - 7xy . 3y + 7xy . 4xy = 7xy(2x - 3y + 4xy)

d) 2525x(y - 1) - 2525y(y - 1) = 2525(y - 1)(x - y)

e) 10x(x - y) - 8y(y - x) =10x(x - y) - 8y[-(x - y)]

                                    = 10x(x - y) + 8y(x - y)

                                    = 2(x - y)(5x + 4y)

oooooooooooooooooooooooooooooooooooooooooooooooolllllllllllllllllllllllllllllllllllllleeeeeeeeeeeeeeeeeeeeellllllllllllllllllllllllllleeeeeeeeeeeeeee

đây nè

k mik đi

1=20/20=1/20+4/20+5/20+10/20=1/20+1/5+1/4+1/2

vậy a=20 ; b=5 ; c=4 ; d=2 hoặc chọn lựa thay đổi thứ tự các số miễn là 4 số 2; 4 ; 5 ; 20