Question

So sánh A=1/3 + 1/6 + 1/10 +...+1/561 với phân số 8/9

Answer 1

\(A=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{561}\)

\(A=\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{1122}\)

\(A=\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{33.34}\)

\(A=2.\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{33.34}\right)\)

\(A=2.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{33}-\frac{1}{34}\right)\)

\(A=2.\left(\frac{1}{2}-\frac{1}{34}\right)\)

\(A=2.\left(\frac{17-1}{34}\right)\)

\(A=2.\frac{8}{17}\)

\(A=\frac{16}{17}>\frac{16}{18}=\frac{8}{9}\)

\(\Rightarrow A>\frac{8}{9}\)