Question

[Đức Linh Nguyễn]

So sánh A=1/5+1/13+1/25+1/37+1/61+1/77+1/113 và B=1/2

Answer 1

[Lê Minh Vũ]

Ta có: \(\dfrac{1}{5}+\dfrac{1}{13}+\dfrac{1}{25}+\dfrac{1}{37}+\dfrac{1}{61}+\dfrac{1}{77}+\dfrac{1}{113}\)

=> \(A=\dfrac{1}{5}+A1\left(\dfrac{1}{13}+\dfrac{1}{25}+\dfrac{1}{37}\right)+A2\left(\dfrac{1}{61}+\dfrac{1}{77}+\dfrac{1}{113}\right)\)

Xét: A1: \(\dfrac{1}{13}< \dfrac{1}{12};\dfrac{1}{25}< \dfrac{1}{12};\dfrac{1}{37}< \dfrac{1}{12}\)

=> \(A1< 3\times\dfrac{1}{12}=\dfrac{1}{4}\)

Xét A2: \(\dfrac{1}{61}< \dfrac{1}{60};\dfrac{1}{77}< \dfrac{1}{60};\dfrac{1}{113}< \dfrac{1}{60}\)

=> \(A2< 3\times\dfrac{1}{60}=\dfrac{1}{20}\)

=> \(A=\dfrac{1}{5}+A1+A2< \dfrac{1}{5}+\dfrac{1}{4}+\dfrac{1}{20}=\dfrac{1}{2}\)

=> \(A< \dfrac{1}{2}\)

=> \(A< B\)