Question

So sánh :

\(A=\sqrt{2}+\sqrt{6}+\sqrt{12}+\sqrt{20}+\sqrt{30}+\sqrt{42}\) và \(B=24\)

Answer 1

Ta có

\(\sqrt{2}\)=\(\sqrt{\dfrac{8}{4}}\)<\(\sqrt{\dfrac{9}{4}}\)=\(\dfrac{3}{2}\)

\(\sqrt{6}\)=\(\sqrt{\dfrac{24}{4}}\)<\(\sqrt{\dfrac{25}{4}}\)=\(\dfrac{5}{2}\)

\(\sqrt{12}\)=\(\sqrt{\dfrac{48}{4}}\)<\(\sqrt{\dfrac{49}{4}}\)=\(\dfrac{7}{2}\)

\(\sqrt{20}\)=\(\sqrt{\dfrac{80}{4}}\)<\(\sqrt{\dfrac{81}{4}}\)=\(\dfrac{9}{4}\)

\(\sqrt{30}\)=\(\sqrt{\dfrac{120}{4}}\)<\(\sqrt{\dfrac{121}{4}}\)=\(\dfrac{11}{2}\)

\(\sqrt{42}\)=\(\sqrt{\dfrac{168}{4}}\)<\(\sqrt{\dfrac{169}{4}}\)=\(\dfrac{13}{2}\)

Do đó A<\(\dfrac{3}{2}+\dfrac{5}{2}+\dfrac{7}{2}+\dfrac{9}{2}+\dfrac{11}{2}+\dfrac{13}{2}\)=24

Vậy A<24