Question

[Đinh Thị Thanh Thủy]

Bài 1 So Sánh S với 2 biết: 

S= 1 + 1/3 + 1/6 + 1/10 + ... + 1/45

Answer 1

[Isolde Moria]

Ta có

\(\frac{S}{2}=2+\frac{1}{6}+\frac{1}{12}+.....+\frac{1}{90}\)

\(\Rightarrow\frac{S}{2}=2+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{9}-\frac{1}{10}\)

\(\Rightarrow\frac{S}{2}=2+\frac{1}{2}-\frac{1}{10}\)

\(\Rightarrow\frac{S}{2}=2+\frac{2}{5}=\frac{12}{5}\)

\(\Rightarrow\frac{S}{2}=\frac{6}{5}< 2\)

=> S<2

Answer 2

[Lê Nguyên Hạo]

\(S=1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\)

\(S=\frac{2}{2}+\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{90}\)

\(S=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{9.10}\)

\(S=2\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{9.10}\right)\)

\(S=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\right)\)

\(S=2\left(1-\frac{1}{10}\right)\)

có : \(1-\frac{1}{10}< 1\Rightarrow2\left(1-\frac{1}{10}\right)< 2.1\)

Vậy: \(S< 2\)