Question

Cho P = \(\dfrac{12}{1.4.7}+\dfrac{12}{7.4.10}+\dfrac{12}{7.10.13}+...+\dfrac{12}{54.57.60}.\) Chứng minh: P < \(\dfrac{1}{2}\).

Answer 1

\(P=\dfrac{12}{1\cdot4\cdot7}+\dfrac{12}{4\cdot7\cdot10}+\dfrac{12}{7\cdot10\cdot13}+...+\dfrac{12}{54\cdot57\cdot60}\)

\(P=\dfrac{12}{6}\left(\dfrac{1}{1\cdot4}-\dfrac{1}{4\cdot7}+\dfrac{1}{4\cdot7}-\dfrac{1}{7\cdot10}+...+\dfrac{1}{54\cdot57}-\dfrac{1}{57\cdot60}\right)\)

\(P=2\left(\dfrac{1}{1\cdot4}-\dfrac{1}{57\cdot60}\right)\)

\(P=\dfrac{2}{4}-\dfrac{2}{57\cdot60}=\dfrac{1}{2}-\dfrac{1}{57\cdot30}\)

\(\Rightarrow P< \dfrac{1}{2}\)