Question

Cho \(\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\)

CM: \(a^3+b^3+c^3⋮3\)

Answer 1

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

<=> \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ac}=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\)

<=> \(2\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}\right)=0\)

<=> \(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}=0\)

<=>\(\dfrac{a+b+c}{abc}=0\)

=> a+b+c =0

=> a³+b³+c³=3abc ⋮3 (bn tự cm)

=> a³+b³+c³ ⋮3 (đpcm)