Question

[Vũ Nhật Minh]

Cho A=1/11+1/12+1/13+1/14+...+1/70. Chứng minh:

a)A>4/3

b)A<2,5

Answer 1

[Đức Nguyễn]

\(A=\frac{1}{11}+\frac{1}{12}+...+\frac{1}{70}\)

\(A=\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}\right)+\left(\frac{1}{21}+\frac{1}{22}+...+\frac{1}{30}\right)\)

\(+\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}\right)\)

\(+\left(\frac{1}{61}+\frac{1}{62}+...+\frac{1}{70}\right)\)

\(\Rightarrow A< \frac{1}{10}\cdot10+\frac{1}{20}\cdot10+\frac{1}{30}\cdot10+...+\frac{1}{60}\cdot10\)

\(A< 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{6}\)

\(A< 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{6}+\left(\frac{1}{4}+\frac{1}{5}\right)\)

\(A< 2+0,45< 2,5\)

Answer 2

[Đức Nguyễn]

\(A=\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{70}\)

\(A>\left(\frac{1}{20}+\frac{1}{20}+..+\frac{1}{20}\right)+\left(\frac{1}{30}+...+\frac{1}{30}\right)+...+\left(\frac{1}{70}+\frac{1}{70}+...+\frac{1}{70}\right)\)

\(A>\frac{1}{2}+\frac{1}{3}+..+\frac{1}{7}\)

\(A>\frac{223}{140}>\frac{4}{3}\)