S:E15.208

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.

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\in \mathbb{N}} 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 $a=1006$. Deci avem suma de două numere consecutive

$$1007+1008=2015.$$

Pentru $N=5$ se obține $2a+6=806$, cu $a=400$. Deci avem suma de $5$ numere consecutive

$$401+402+403+404+405=2015.$$

Pentru $N=10$ se obține $2a+11=403$, cu $a=196$. Deci avem suma de $10$ numere consecutive

$$197+198+\ldots +206=2015.$$

Pentru $N=13$ se obține $2a+14=310$, cu $a=148$. Deci avem suma cu $13$ termeni, numere consecutive

$$149+150+\ldots +161=2015.$$

Pentru $N=26$ se obține $2a+27=155$, cu $a=64$. Deci avem suma de $26$ de numere consecutive

$$65+66+\ldots +67=2015.$$

Pentru $N=31$ se obține $2a+32=130$, cu $a=49$. Deci avem suma de $31$ de numere consecutive

$$50+51+\ldots +80=2015.$$

Pentru $N=62$ se obține $2a+63=65$, 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