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'''E:7456 (Mirela-Petrina Timiș)''' | '''E:7456 (Mirela-Petrina Timiș)''' | ||
''Să se arate că fracția <math>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} </math> se simlifică prin <math>16</math>'' | ''Să se arate că fracția <math>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} </math> se simlifică prin <math>16</math>.'' | ||
'''Soluție''' | '''Soluție''' | ||
Pentru orice <math>F = \frac{23^{n | Pentru orice <math>n\in \mathbb{N}^\ast</math> avem <math>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)} </math>. | ||
Pentru orice numere reale <math>a,b \in \mathbb{R}</math> și <math>n\in \mathbb{N}^\ast</math>, are loc egalitatea <math>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)</math>, deci | |||
oricare ar fi numerele naturale <math>a,b \in \mathbb{N}</math>, cu <math>a>b</math>, și oricare ar fi <math>n\in \mathbb{N}^\ast</math>, există <math>M\in \mathbb{N}</math> astfel încât <math>a^n - b^n = M\cdot \left(a-b\right)</math>. | |||
Atunci există numerele naturale <math>M_1,M_2,M_3 \in \mathbb{N}</math> pentru care <math>23^n-7^n = 16 \cdot M_1</math>, <math>19^n-3^n = 16 \cdot M_2</math>, respectiv <math>41^n-25^n = 16 \cdot M_3</math>. | |||
Deci, pentru orice <math>n\in \mathbb{N}^\ast</math> avem <math>F = \frac{16\cdot \left( 23 \cdot M_1 - 4^{n+1} \cdot M_2 \right)}{41\cdot 16 \cdot M_3} </math>, de unde se poate deduce că fracția <math>F </math> se poate simplifca prin ''<math>16</math>.'' | |||
Revision as of 13:42, 5 January 2025
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 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\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 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{R}} și 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} , are loc egalitatea 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 = \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} .