Question

Cho các số 0 <\(a_1< a_2< a_3< ...< a_{15}\). Chứng minh rằng \(\dfrac{a_1+a_2+a_3+...+a_{15}}{a_5+a_{10}+a_{15}}< 5\)

Answer 1

Vì \(a_1< a_2< a_3< ...< a_{15}\) ta có:

\(\dfrac{a_1+a_2+a_3+...+a_{15}}{a_5+a_{10}+a_{15}}< \dfrac{a_5+a_{10}+a_{15}+a_5+a_{10}+a_{15}+...+a_5+a_{10}+a_{15}}{a_5+a_{10}+a_{15}}\)\(\Rightarrow\dfrac{a_1+a_2+a_3+...+a_{15}}{a_5+a_{10}+a_{15}}< \dfrac{5\left(a_5+a_{10}+a_{15}\right)}{a_5+a_{20}+a_{15}}\)

\(\Rightarrow\dfrac{a_1+a_2+a_3+...+a_{15}}{a_5+a_{10}+a_{15}}< 5\)

\(\rightarrowđpcm\)