Question

aratati ca nr. 2005+2x(1+2+3+......+2004) este patrat perfect

Answer 1

1 + 2 +3 + ... + 2004 = ( 1 +2004 ) · 2004 :2 =2005 ·1002 
2005 + 2 · 2005 ·1002 = 2005 · ( 1 + 2·1002) = 2005 · ( 1 + 2004) 
= 2005 ·2005 =2005²

Answer 2

$\displaystyle 2005+2 \cdot (1+2+3+...+2004)=2005+2 \cdot \frac{2004(2004+1)}{2} = \\ \\ =2005 +2 \cdot \frac{2004 \cdot 2005}{2} =2005+\not2 \cdot \frac{4018020}{\not 2} =2005+4018020= \\ \\ =2005(1+2004)=2005 \cdot 2005=2005^2-p.p$