14183

14183 (Gheorghe Szőllőssy)

Să se calculeze suma $S=\displaystyle \sum _{k=0}^{n}\left(k+1\right)^{2}C_{n}^{k}$.

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 egalitățile

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(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}}

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(2,n) = \displaystyle \sum_{k=0}^n k^2C_n^k = n\left(n+1\right)2^{n-2}} .

Cum $S=\displaystyle \sum _{k=0}^{n}\left(k+1\right)^{2}C_{n}^{k}=\displaystyle \sum _{k=0}^{n}k^{2}C_{n}^{k}+2\cdot \displaystyle \sum _{k=0}^{n}kC_{n}^{k}+\displaystyle \sum _{k=0}^{n}C_{n}^{k}$, se obține $S=n\left(n+1\right)2^{n-2}+2\cdot n2^{n-1}+2^{n}$, deci $S=\displaystyle \sum _{k=0}^{n}\left(k+1\right)^{2}C_{n}^{k}=2^{n-2}\left(n^{2}+5n+4\right)$.