E:7456

E:7456 (Mirela-Petrina Timiș)

Să se arate că fracția 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 F = \frac{23^{n+1}-4^{n+1}\cdot 19^n - 7^n \cdot 23 + 2^{2n+2}\cdot 3^n}{41^{n+1}-5^{2n}\cdot 41} } se simlifică prin 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 16} .

Soluție

Pentru orice 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\in \mathbb{N}^\ast} avem $F={\frac {23\left(23^{n}-7^{n}\right)-4^{n+1}\cdot \left(19^{n}-3^{n}\right)}{41\left(41^{n}-25^{n}\right)}}$.

Pentru orice numere reale $a,b\in \mathbb {R}$ și $n\in \mathbb {N} ^{\ast }$, are loc egalitatea $a^{n}-b^{n}=\left(a-b\right)\cdot \left(a^{n-1}+a^{n-2}b+a^{n-3}b^{2}+\ldots +a^{2}b^{n-3}+ab^{n-2}+b^{n-1}\right)$, deci

oricare ar fi numerele naturale 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,b \in \mathbb{N}} , 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 a>b} , și oricare ar fi 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\in \mathbb{N}^\ast} , există 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 M\in \mathbb{N}} astfel încât 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 - b^n = M\cdot \left(a-b\right)} .

Atunci există numerele naturale 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 M_1,M_2,M_3 \in \mathbb{N}} pentru care 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 23^n-7^n = 16 \cdot M_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 19^n-3^n = 16 \cdot M_2} , respectiv 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 41^n-25^n = 16 \cdot M_3} .

Deci, pentru orice 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\in \mathbb{N}^\ast} avem 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 F = \frac{16\cdot \left( 23 \cdot M_1 - 4^{n+1} \cdot M_2 \right)}{41\cdot 16 \cdot M_3} } , de unde se poate deduce că fracția 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 F } se poate simplifca prin 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 16} .