Shortlist 18

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Antonache Emanuel
Euclid
Posts: 37
Joined: Sat Feb 28, 2009 4:15 pm
Location: Targoviste, Dambovita

Shortlist 18

Post by Antonache Emanuel »

Fie \( a\in\mathbb{N}, a\ge2 \). Definitm sirul \( (a_n)_{n\ge0} \) prin relatiile: \( x_0=\frac{a^2}{4}, x_1=\frac{2a^4-4a^3-a^2+4a}{4} \) si \( x_{n+1}-(4a^2-2)x_n+x_{n-1}=0 \), pentru \( n\ge1 \). Sa se arate ca numarul \( 2x_n-\frac{a^2-2}{2} \) este patrat perfect pentru orice \( n\in\mathbb{N} \).
Stefan Alexe, Pitesti, Shortlist 2006
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