27020

27020 (Gheorghe Szöllösy)

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\lfloor\frac{n}{2}\right.} \frac{1}{4^k \cdot (k!)^2 \cdot (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

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

Avem $a_{n}=\left({\frac {1}{2^{n}}}\right)C_{2}n^{n}$, iar pe de altă parte,

$$a_{n}=C_{n}^{0}\cdot C_{n}^{0}+C_{n}^{1}\cdot C_{(}n-1)^{1}\left.{\frac {1}{4}}\right.+C_{n}^{2}\cdot C_{(}n-2)^{1}\left.{\frac {1}{4^{2}}}\right.+...=$$

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\lfloor\frac{n}{2}\right\rfloor} C_n^k C_(n-k)^k \cdot \left.\frac{1}{4^k}\right. = n! \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor} \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. .}