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 $n$ au loc egalitățile

$$S(0,n)=\displaystyle \sum _{k=0}^{n}C_{n}^{k}=2^{n}$$

$$S(1,n)=\displaystyle \sum _{k=0}^{n}kC_{n}^{k}=n2^{n-1}$$

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

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