Question

Cho \(4\) số khác là \(a_1,a_2,a_3,a_4\) thoả mãn a₂² = a₁a₃ ;a₃³ = a₂a₄. Chứng tỏ 

Answer 1

a2^2=a1*a3

=>a1/a2=a2/a3

 

a3^2=a2*a4

=>a2/a3=a3/a4

=>a1/a2=a2/a3=a3/a4=k

=>a1=a2k; a2=a3*k; a3=a4*k

=>a1=a2*k; a2=a4*k^2; a3=k*a4

=>a1=a4*k^3; a2=a4*k^2; a3=k*a4

\(\dfrac{a_1^3+a_2^3+a_3^3}{a_2^3+a_3^3+a_4^3}=\dfrac{a_4^3\cdot k^9+a_4^3\cdot k^6+a_4^3\cdot k^3}{a_4^3\cdot k^6+a_4^3\cdot k^3+a_4^3}\)

\(=\dfrac{k^3\left(a_4^3\cdot k^6+a_4^3\cdot k^3+a_3^4\right)}{a_4^3\cdot k^6+a_4^3\cdot k^3}=k^3\)

\(\dfrac{a1}{a4}=\dfrac{a_4\cdot k^3}{a_4}=k^3\)

=>\(\dfrac{a_1^3+a_2^3+a_3^3}{a_2^3+a_3^3+a_4^3}=\dfrac{a_1}{a_4}\)