Olimpíadas ⇒ Ache uma expressão Tópico resolvido

[RonaldoJr]

Não estou conseguindo achar uma "expressão", "valor" para n...
Sendo [tex3]n[/tex3] e [tex3]j \in Z[/tex3]

[tex3]\left( j- \frac{1}{2} \right) ^2 + 2 \leq \left( 2n + \frac{1}{4}\right) \leq \left( j+ \frac{1}{2} \right)^2[/tex3]

Vlw

[jrneliodias]

Olá Ronaldo Jr

[tex3]\left( j- \frac{1}{2} \right) ^2 + 2 \leq \left( 2n + \frac{1}{4}\right) \leq \left( j+ \frac{1}{2} \right)^2[/tex3]

[tex3]\left( j- \frac{1}{2} \right) ^2 + 2-\frac{1}{4}\,\,\leq\,\,2n\,\,\leq\,\,\left( j+ \frac{1}{2} \right)^2-\frac{1}{4}[/tex3]

[tex3]\left( j- \frac{1}{2} \right) ^2 +\frac{7}{4}\,\,\leq\,\,2n\,\,\leq\,\,\left( j+ \frac{1}{2} \right)^2-\frac{1}{4}[/tex3]

[tex3]\frac{1}{2}\cdot\left[\left( j- \frac{1}{2} \right) ^2 +\frac{7}{4}\right]\,\,\leq\,\,n\,\,\leq\,\,\frac{1}{2}\cdot\left[\left( j+ \frac{1}{2} \right)^2-\frac{1}{4}\right][/tex3]

[tex3]\frac{1}{2}\cdot\left[\left( j- \frac{1}{2} \right) ^2\right] +\frac{7}{8}\,\,\leq\,\,n\,\,\leq\,\,\frac{1}{2}\cdot\left[\left( j+ \frac{1}{2} \right)^2\right]-\frac{1}{8}[/tex3]

[tex3]\frac{4j^2-4j+8}{8}+\frac{7}{8}\,\,\leq\,\,n\,\,\leq\,\,\frac{4j^2+4j+1}{8}-\frac{1}{8}[/tex3]

[tex3]\boxed{\frac{(j-2)(j-1)}{2}\,\,\leq\,\,n\,\,\leq\,\,\frac{j(j-1)}{2}}[/tex3]

Espero que tenha ajudado, abraço