S:E15.208: Difference between revisions

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''Determinați toate numerele naturale consecutive care au suma <math>2015</math>.''
''Determinați toate numerele naturale consecutive care au suma <math>2015</math>.''


'''Soluția 1'''
'''Soluția 1.''' (aici poate fi consultată o altă soluție '''[[S:E15.208-sol2]])'''
 
Fie <math>a\in \mathbb{N}</math> și <math>N\in \mathbb{N}\setminus\left\{0,1\right\}</math> numere naturale pentru care <math display="block">\left(a+1\right) + \left(a+2\right)+\ldots+\left(a+N\right)=2015.</math>
Fie <math>a\in \mathbb{N}</math> și <math>N\in \mathbb{N}\setminus\left\{0,1\right\}</math> numere naturale pentru care <math display="block">\left(a+1\right) + \left(a+2\right)+\ldots+\left(a+N\right)=2015.</math>
În mod echivalent, se obține <math display="block">Na+\left(1+2+\ldots+N\right)=2015,</math> deci
În mod echivalent, se obține <math display="block">Na+\left(1+2+\ldots+N\right)=2015,</math> deci
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N\cdot\left(2a+1+N\right) = 2\cdot2015.
N\cdot\left(2a+1+N\right) = 2\cdot2015.
</math>
</math>
Din <math>a\ge 0</math>, avem <math>N\left(N+1\right)\le N\left(2a+1+N\right) = 2024</math>. Cum <math>63\cdot 64 \le 4030 \le 64\cdot 65</math>, se deduce că <math>N\le 63</math>.  
Din <math>a\ge 0</math>, avem <math>N\left(N+1\right)\le N\left(2a+1+N\right) = 2024</math>. Cum <math>63\cdot 64 \le 4030 \le 64\cdot 65</math>, se deduce că <math>N\le 63</math>.  


Din <math>N | 4030</math> și  <math>N\le 63</math> rezultă <math>N\in \left\{2,5,10,13,26,31, 62 \right\}</math>.
Din <math>N | 4030</math> și  <math>N\le 63</math> rezultă <math>N\in \left\{2,5,10,13,26,31, 62 \right\}</math>.
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</math>
</math>


'''Soluția 2'''
Aici poate fi consultată o altă soluție [[S:E15.208-sol2]]
 
Fie <math>N\in \mathbb{N}\setminus\left\{0,1\right\}</math> numărul de termeni ai sumei.
 
Cum suma a <math>4n</math> numere consecutive este un număr par, iar <math>2015</math> este număr impar, deducem că <math>4 \nmid N</math>.
 
Pentru <math>N=4n+2</math>, cu <math>n\in\mathbb{N}</math>, suma se poate scrie
<math\left(b-2n\right)+\left(b-2n+1\right)+\ldots+\left(b-1\right) + b + \left(b+1\right)+\left(b+1\right)+\ldots+\left(b+2n\right) + \left(a+2n+1\right)=2015,</math>  unde <math>b\in \mathbb{N}</math>, cu <math>b\ge 2n</math>. Se obține
<math>\left(2n+1\right)\left(2b+1\right)=1\cdot 5 \cdot 13 \cdot 31.</math>
 
Pentru <math>2n+1=1</math> se obține <math>b=1007</math> și suma \eqref{eq2cls6}
 
Pentru <math>2n+1=5</math> se obține <math>b=201</math> și suma \eqref{eq4cls6}.
 
Pentru <math>2n+1=13</math> se obține <math>b=77</math> și suma \eqref{eq6cls6}.
 
Pentru <math>2n+1=31</math> se obține <math>b=32</math> și suma \eqref{eq7cls6}.
 
Celelalte situații posibile nu satisfac condiția <math>b\ge 2n<math>.
 
Pentru <math>N=2n+1</math>, cu <math>n\in \mathbb{N}^\ast</math>, suma se poate scrie
<math>\left(b-n\right)+\left(b-n+1\right)+\ldots+\left(b-1\right)+b+\left(b+1\right)+\left(b+2\right)+\ldots+\left(b+n\right)=2015,</math> unde <math>b\in \mathbb{N}</math>, cu <math>b\ge n</math>.
 
Se obține <math>\left(2n+1\right)\cdot b = 5 \cdot 13\cdot 31.</math>
 
Pentru <math>2n+1=5</math> se obține <math>b=403</math> și suma \eqref{eq3cls6}.
 
Pentru <math>2n+1=13</math> se obține <math>b=155</math> și suma \eqref{eq5cls6}.
 
Pentru <math>2n+1=31</math> se obține <math>b=65</math> și suma \eqref{eq8cls6}.
 
Celelalte situații posibile nu satisfac condiția <math>b\ge n</math>.

Latest revision as of 17:10, 20 August 2025

S:E15.208 (Angela Lopată)

Determinați toate numerele naturale consecutive care au 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 2015} .

Soluția 1. (aici poate fi consultată o altă soluție S:E15.208-sol2)

Fie 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}\setminus\left\{0,1\right\}} numere naturale 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 \left(a+1\right) + \left(a+2\right)+\ldots+\left(a+N\right)=2015.} În mod echivalent, 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 Na+\left(1+2+\ldots+N\right)=2015,} 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 N\cdot\left(2a+1+N\right) = 2\cdot2015. } Din 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\ge 0} , 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 N\left(N+1\right)\le N\left(2a+1+N\right) = 2024} . Cum 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 63\cdot 64 \le 4030 \le 64\cdot 65} , se deduce că 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\le 63} .

Din 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 | 4030} 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\le 63} rezultă 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 \left\{2,5,10,13,26,31, 62 \right\}} .

Pentru 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=2} 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 2a+3 = 2015} , 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 =1006} . Deci avem suma de două numere consecutive 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 1007+1008=2015. }

Pentru 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=5} 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 2a+6 = 806} , 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 = 400} . Deci avem suma de 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 5} numere consecutive 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 401+402+403+404+405=2015. }

Pentru 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=10} 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 2a+11 = 403} , 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 = 196} . Deci avem suma de 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 10} numere consecutive 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 197+198+\ldots+206=2015. }

Pentru 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=13} 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 2a+14 = 310} , 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 = 148} . Deci avem suma 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 13} termeni, numere consecutive 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 149+150+\ldots+161=2015. }

Pentru 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=26} 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 2a+27 = 155} , 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 = 64} . Deci avem suma de 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 26} de numere consecutive 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 65+66+\ldots+67=2015. }

Pentru 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=31} 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 2a+32 = 130} , 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 = 49} . Deci avem suma de 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 31} de numere consecutive 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 50+51+\ldots+80=2015. }

Pentru 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=62} 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 2a+63 = 65} , 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 = 1} . Deci avem suma de 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 62 } de numere consecutive 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 2+3+\ldots+63=2015. }

Aici poate fi consultată o altă soluție S:E15.208-sol2

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