a:b=b:c=c:a=>a/b=b/c=c/a=a+b+c/b+c+a=1

suy ra: a/b=1 suy ra: a=b

b/c=1 =>b=c

suy ra: a=b=c

suy ra: a^2.b^2.c^1930:b^1935=1.1.1:1=1

Ta có:

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

\(\Rightarrow a=b=c\)

\(\Rightarrow M=\frac{a^3.b^2.c^{1930}}{b^{1935}}=\frac{b^{1935}}{b^{1935}}=1\)

Áp dụng tỉ dãy số bằng nhau:

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

Khi đó: \(\frac{a^3.b^2.c^{1930}}{b^{1935}}\Leftrightarrow\frac{b^{1935}}{b^{1935}}=b^{1935}:b^{1935}=1\)

theo bài ra và theo tc của dãy tỉ số bằng nhau ta có :a/b=b/c=c/a suy ra a+b+c/b+c+a=1

suy ra a=b=c suy ra a^3*b^2*c^1930=b^1935 suyra b^1935/b^1935=1

Vì \(a+b+c\ne0\)\(\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{a}=\frac{a+b+c}{b+c+a}=1\)

\(\Rightarrow a=b=c\)\(\Rightarrow M=\frac{a^3.b^2.c^{1930}}{b^{1935}}=\frac{b^3.b^2.b^{1930}}{b^{1935}}=\frac{b^{3+2+1930}}{b^{1935}}=\frac{b^{1935}}{b^{1935}}=1\)

Vậy \(M=1\)

Ta có :\(\frac{a}{b}=\frac{b}{c}=\frac{c}{a}=\frac{a+b+c}{b+c+a}=1\Rightarrow a=b=c\Rightarrow\frac{a^3.b^2.c^{1930}}{b^{1935}}=\frac{b^3.b^{1930}}{b^{1933}}=1\)