Đặt :
\(A=\)\(\dfrac{1}{5}+\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}+\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}\)
\(A=\dfrac{1}{5}+\left(\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}\right)+\left(\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}\right)\)
Ta thấy :
\(\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}< \dfrac{1}{12}+\dfrac{1}{12}+\dfrac{1}{12}\)
\(\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}< \dfrac{1}{60}+\dfrac{1}{61}+\dfrac{1}{62}\)
\(\Rightarrow A< \dfrac{1}{5}+\left(\dfrac{1}{12}+\dfrac{1}{12}+\dfrac{1}{12}\right)+\left(\dfrac{1}{60}+\dfrac{1}{60}+\dfrac{1}{60}\right)\)
\(\Rightarrow A< \dfrac{1}{5}+\dfrac{1}{12}.3+\dfrac{1}{60}.3\)
\(\Rightarrow A< \dfrac{1}{5}+\dfrac{1}{4}+\dfrac{1}{20}\)
\(\Rightarrow A< \dfrac{10}{20}=\dfrac{1}{2}\)
\(\Rightarrow A< \dfrac{1}{2}\rightarrowđpcm\)
