Question

Vă rog frumos să mă ajutați!​

Answer 1

Explicație pas cu pas:

$S_{2} = -2+4-6+8-...+2004-2006+2008-2010 = \\ = -2 \cdot (-1+2-3+4-...+1002-1003+1004-1005) \\ = -2 \cdot \Big[(1-2)+(3-4)+(5-6)+...+(2009-2010)\Big] \\ = -2 \cdot (\underbrace{-1-1-1-...-1}_{1005} = -2 \cdot (-1005) = \bf 2010$

$S_{3} = +1+2+3+4-(+1+2+3+4) = 10 - 10 = \bf 0$

$S_{4} = (1-2+3-4+5-6+...+2009-2010) \cdot 2+2010 = \\ = \Big[(1-2)+(3-4)+(5-6)+...+(209-2010)\Big] \cdot 2+2010 \\ = (\underbrace{-1-1-1-...-1}_{1005} \cdot 2+2010 = -1005 \cdot 2+2010 \\ = -2010+2010 = \bf 0$

$S_{5} = -5-10-15-...-2005-2010+(1+2+3+...+402) = \\ = -5 \cdot (1+2+3+...+402) + (1+2+3+...+402) \\ = (1+2+3+...+402) \cdot (-5+1) = -4 \cdot \frac{402 \cdot 403}{2} \\ = -4 \cdot 201 \cdot 403 = \bf -324012$

$S_{6} = -5-10-15-...-2005-2010 = \\ = -5 \cdot (1+2+3+...+402) \\ = -5 \cdot \frac{402 \cdot 403}{2} = -5 \cdot 201 \cdot 403 = \bf -405015$