Question

Să se arate că:

$\it \dfrac{1}{2}\cdot\dfrac{3}{4}\cdot\dfrac{5}{6}\cdot\ ...\ \cdot\dfrac{99}{100}\ \textless \ \dfrac{1}{10}$

Mulțumesc !!!

Answer 1

$\text{Fie }P = \dfrac{1}{2}\cdot \dfrac{3}{4}\cdot \dfrac{5}{6}\cdot...\cdot\dfrac{97}{98}\cdot \dfrac{99}{100},\quad Q =\dfrac{2}{3}\cdot \dfrac{4}{5}\cdot \dfrac{6}{7}\cdot...\cdot \dfrac{98}{99}\cdot \dfrac{100}{100} \\ \\ P\cdot Q = \dfrac{1}{2}\cdot \dfrac{2}{3}\cdot \dfrac{3}{4}\cdot \dfrac{4}{5}\cdot .. .\cdot \dfrac{98}{99}\cdot \dfrac{99}{100} = \dfrac{1}{100} \\ \\ \dfrac{1}{2}<\dfrac{2}{3},~\dfrac{3}{4}<\dfrac{4}{5},~...~,\dfrac{97}{98}<\dfrac{98}{99},\dfrac{99}{100}<\dfrac{100}{100} \\ \\ \\ \Rightarrow P<Q \Big|\cdot P$

$\Rightarrow P^2 < P\cdot Q \\ \\ \Rightarrow P^2 < \dfrac{1}{100} \\ \\ \Rightarrow P < \dfrac{1}{10}$