Question

Chứng minh rằng:\(\frac{1}{3}+\frac{1}{31}+\frac{1}{36}+\frac{1}{37}+\frac{1}{47}+\frac{1}{53}+\frac{1}{61}<\frac{1}{2}\)

 

Answer 1

\(\frac{1}{3}+\frac{1}{31}+\frac{1}{36}+\frac{1}{37}+\frac{1}{47}+\frac{1}{53}+\frac{1}{61}

Answer 2

Ta có : \(\frac{1}{3}+\frac{1}{31}+\frac{1}{36}+\frac{1}{37}+\frac{1}{47}+\frac{1}{53}+\frac{1}{61}< \frac{1}{3}+\left(\frac{1}{30}+\frac{1}{30}+\frac{1}{30}\right)+\)\(\left(\frac{1}{45}+\frac{1}{45}+\frac{1}{45}\right)\)

\(=\frac{1}{3}+\frac{3}{30}+\frac{3}{45}=\frac{1}{2}\)

\(\Rightarrow\)\(\frac{1}{3}+\frac{1}{31}+\frac{1}{36}+\frac{1}{37}+\frac{1}{47}+\frac{1}{53}+\frac{1}{61}< \frac{1}{2}\)

\(\RightarrowĐPCM\)

Answer 3

\(\frac{1}{3}+\frac{1}{31}+\frac{1}{36}+\frac{1}{37}+\frac{1}{47}+\frac{1}{53}+\frac{1}{61}< \frac{1}{3}+\frac{1}{31}+\frac{1}{31}+\frac{1}{31}+\frac{1}{47}+\frac{1}{47}+\frac{1}{47}\)

\(=\frac{1}{3}+3.\frac{1}{31}+3.\frac{1}{47}=\frac{1}{3}+\frac{3}{31}+\frac{3}{47}=\frac{2159}{4371}< \frac{1}{2}\left(đpcm\right).\)