16402

16402 (Cristina Vijdeluc, Salonic și Mihai Vijdeluc, Baia Mare)

Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}^*} și numerele pozitive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1, x_2, \dots, x_n} care verifică relația

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n} = \frac{n}{n+1}.}

Arătați că

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x_1 + 2} + \frac{1}{x_2 + 6} + \dots + \frac{1}{x_n + n(n+1)} \leq \frac{n}{2(n+1)}.}

Soluție:

Din inegalitatea mediilor armonică și aritmetică, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{a+b} \leq \frac{1}{4}\left(\frac{1}{a} + \frac{1}{b}\right),} pentru oricare Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b > 0.} Aplicând succesiv această inegalitate obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x_1 + 2} + \frac{1}{x_2 + 6} + \dots + \frac{1}{x_n + n(n+1)} \leq \frac{1}{4} \left( \frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n} + \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n(n+1)} \right) =} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4} \left( \frac{n}{n+1} + 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \dots + \frac{1}{n} - \frac{1}{n+1} \right) = \frac{n}{2(n+1)}.}