14183

14183 (Gheorghe Szőllőssy)

Să se calculeze 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 = \displaystyle \sum_{k=0}^n \left(k+1\right)^2C_n^k} .

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 p} considerăm 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(p,n) = \displaystyle \sum_{k=0}^n k^pC_n^k} .

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} au loc egalitățileFailed 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(0,n) = \displaystyle \sum_{k=0}^n C_n^k = 2^n} 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(1,n) = \displaystyle \sum_{k=0}^n kC_n^k = n2^{n-1}}

$$S(2,n)=\displaystyle \sum _{k=0}^{n}k^{2}C_{n}^{k}=n\left(n+1\right)2^{n-2}$$

Cum 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 = \displaystyle \sum_{k=0}^n \left(k+1\right)^2C_n^k = \displaystyle \sum_{k=0}^n k^2C_n^k + 2\cdot \displaystyle \sum_{k=0}^n kC_n^k + \displaystyle \sum_{k=0}^n C_n^k } , se obține 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\left(n+1\right)2^{n-2} + 2\cdot n2^{n-1} + 2^n,} deci 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 = \displaystyle \sum_{k=0}^n \left(k+1\right)^2C_n^k = 2^{n-2}\left(n^2+5n+4\right).}