23964

23964 (Marin Bancoș)

Să de demonstreze inegalitatea 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 \sum_{i=2}^{n} \sqrt[i]{\left(i!\right)^2} < \frac{2n^3+9n^2+13n-24}{24} .}

Soluție

Pentru orice număr natural 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} , cu 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\ge 2} are loc inegalitatea 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 \sqrt[n]{n!} = \sqrt[n]{1\cdot 2\cdot \ldots \cdot n} < \frac{1+2+\ldots+n}{n} = \frac{n+1}{2}.} AtunciFailed 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 \sum_{i=2}^{n} \sqrt[i]{\left(i!\right)^2} < \sum_{i=2}^{n} \frac{\left(i+1\right)^2}{4}} Suma 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 S_n=\sum_{i=2}^{n} \frac{\left(i+1\right)^2}{4}} poate fi calculată prin mai multe metode. De exemplu, avemFailed 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 S_n = \frac{1}{4} \cdot \sum_{i=3}^{n+1} i^2 = \frac{1}{4} \left(\frac{n\left(n+1\right)\left(2n+1\right)}{6} - 5\right) = \frac{2n^3+9n^2+13n-24}{24}.} În concluzie, pentru orice 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\ge 2 } are loc inegalitateaFailed 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 \sum_{i=2}^{n} \sqrt[i]{\left(i!\right)^2} < \frac{2n^3+9n^2+13n-24}{24} .} ◎ soluție oferită de D. Bărbosu