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- Ta có: \(\dfrac{a}{b}< \dfrac{c}{d}\) (gt)

=>\(ad< bc\) 

=>\(ad+ab< bc+ab\)

=>\(a\left(b+d\right)< b\left(a+c\right)\)

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

- Ta có: \(\dfrac{c}{d}>\dfrac{a}{b}\) (gt)

=>\(bc>ad\)

=>\(bc+cd>ad+cd\)

=>\(c\left(b+d\right)>d\left(a+c\right)\)

=>\(\dfrac{c}{d}>\dfrac{a+c}{b+d}\) (2)

- Từ (1) và (2) suy ra: \(\dfrac{a}{b}< \dfrac{a+c}{b+d}< \dfrac{c}{d}\)

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

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

Tương tự ta chứng minh được \(\frac{b}{a+b}+\frac{c}{b+c}+\frac{d}{c+d}+\frac{a}{a+d}>1\)(2)

mà \(\frac{a}{a+b}+\frac{b}{a+b}+\frac{c}{b+c}+\frac{b}{b+c}+\frac{d}{c+d}+\frac{c}{c+d}+\frac{a}{a+d}+\frac{d}{a+d}=4\)(3)

Từ (1) (2) (3) => \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}=2\left(a;b;c;d\inℕ\right)\)

Câu hỏi của CTV - Toán lớp 8 - Học toán với OnlineMath

Đặt A/B=C/D=k

=>A=k*B; C=D*k

A/B=k*B/B=k

\(\dfrac{A+C}{B+D}=\dfrac{k\cdot B+k\cdot D}{B+D}=k\)

=>\(\dfrac{A}{B}=\dfrac{A+C}{B+D}\)

\(b)\)

\(4n-3⋮3n-2\)

\(\Leftrightarrow3\left(4n-3\right)⋮3n-2\)

\(\Leftrightarrow12n-9⋮3n-2\)

\(\Leftrightarrow\left(12n-8\right)-1⋮3n-2\)

\(\Leftrightarrow4\left(3n-2\right)-1⋮3n-2\)

\(\Leftrightarrow1⋮3n-2\)

\(\Leftrightarrow3n-2\inƯ\left(1\right)=\left\{\pm1\right\}\)

\(\Rightarrow3n\in\left\{1;3\right\}\)

Mà: \(3n⋮3\)

\(\Leftrightarrow3n=3\)

\(\Leftrightarrow n=1\)