E.309. Calculați: 1+2−3+4+5−6+7+8−9+…+2005+2006−2007+2008.1+2-3+4+5-6+7+8-9+ \ldots + 2005+2006-2007+2008.1+2−3+4+5−6+7+8−9+…+2005+2006−2007+2008.
Indicații: Grupăm termenii câte 333 și obținem S=(3+6+9+…+2004)+2008.S=(3+6+9 + \ldots + 2004) + 2008.S=(3+6+9+…+2004)+2008.
Răspuns: S=672346.S=672346.S=672346.
S=1+2+3+4+…+2008−2(3+6+9+…+2007)=S=1+2+3+4+\ldots+2008 - 2(3+6+9+ \ldots+2007)=S=1+2+3+4+…+2008−2(3+6+9+…+2007)= =2008⋅2009:2+2⋅3(1+2+3+…+669)==2008 \cdot 2009 : 2 + 2 \cdot 3(1+2+3+\ldots+669)==2008⋅2009:2+2⋅3(1+2+3+…+669)= =1004⋅2009+3⋅669⋅670==1004 \cdot 2009 + 3 \cdot 669 \cdot 670==1004⋅2009+3⋅669⋅670= =2017036−1344690=672346.=2017036-1344690=672346.=2017036−1344690=672346.
Metoda 2: S=(1+2−3)+(4+5−6)+…+(2005+2006−2007)+2008=S=(1+2-3)+(4+5-6) + \ldots + (2005+2006-2007) + 2008 =S=(1+2−3)+(4+5−6)+…+(2005+2006−2007)+2008= =(3+6+9+…+2004)+2008==(3 + 6 + 9 + \ldots + 2004) + 2008==(3+6+9+…+2004)+2008= =3(1+2+3+…+668)+2008==3(1+2+3+ \ldots + 668) + 2008==3(1+2+3+…+668)+2008= =3⋅668⋅669:2+2008=672346.=3 \cdot 668 \cdot 669 : 2 + 2008=672346.=3⋅668⋅669:2+2008=672346.