27020

27020 (Gheorghe Szöllösy)

\textit{italic} 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 \sum_{k=0}^{\left[\frac{n}{2}\right]} \frac{1}{4^k \cdot (k!)^2 (n-2k)!}, \quad n \geq 1. }

Soluție:

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 a_n } coeficientul lui $X^{n}$ din rezolvarea lui

$$P(X)=\left(X+{\frac {1}{2}}\right)^{2n}=\left(X(1+X)+{\frac {1}{4}}\right)^{n}=\sum _{k=0}^{n}C_{n}^{k}X^{n-k}(!-X)^{n-k}\left.{\frac {1}{4^{k}}}\right..$$

Avem $a_{n}=\left.{\frac {1}{2^{n}}}\right.C_{2n}^{n}$, iar pe de altă parte, 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_n = C_n^0 \cdot C_n^0 + C_n^1 \cdot C_{n-1}^1 \left.\frac{1}{4}\right. + C_n^2 \cdot C_{n-2}^1\left.\frac{1}{4^2}\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 = \sum_{k=0}^{\left[\frac{n}{2}\right]} C_n^k C_{n-k}^k \cdot \left.\frac{1}{4^k}\right. = n! \sum_{k=0}^{\left[\frac{n}{2}\right]} \frac{1}{(k!)^2 (n-k)! 4^k},}

deci suma este egală 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 \left.\frac{(2n!)}{2^n(n!)^3}\right. .}