Question

[Đặng Huy Khánh Nam]

So sánh

1/31 + 1/32 +1/33 + ... + 1/89 + 1/90 và 5/6

Answer 1

[Xyz OLM]

Ta có :\(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{90}=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{60}\right)+\left(\frac{1}{61}+\frac{1}{62}+...+\frac{1}{90}\right)\)

                   60 số hạng                                                              30 số hạng                                     30 số hạng

\(>\left(\frac{1}{60}+\frac{1}{60}+...+\frac{1}{60}\right)+\left(\frac{1}{90}+\frac{1}{90}+...+\frac{1}{90}\right)=30.\frac{1}{60}+30.\frac{1}{90}=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\)

Vậy \(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{90}>\frac{5}{6}\)

Answer 2

[.]

Ta có: \(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}>\frac{1}{60}+\frac{1}{60}+...+\frac{1}{60}=30.\frac{1}{60}=\frac{1}{2}\)

Lại có: \(\frac{1}{61}+\frac{1}{62}+\frac{1}{63}+...+\frac{1}{90}>\frac{1}{90}+\frac{1}{90}+...+\frac{1}{90}=30.\frac{1}{90}=\frac{1}{3}\)

\(\Rightarrow\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{90}>\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\)

\(\Rightarrow\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{90}>\frac{5}{6}\) (đpcm)