ex . 4 punctele d si e dau coroana

   
[tex]\displaystyle 4d) \\ S_4=3+5+7+9+\cdots + 2013 -2-4-6-8-\cdots - 2012 \\ \text{Calculam numarul de termeni pozitivi } n_1 .} \\ \\ n_1 = \frac{2013-3}{2} +1 = \frac{2010}{2} +1 = 1005 +1 = 1006\\ \\ \text{Calculam numarul de termeni negativi } n_2 .} \\ \\ n_2 = \frac{2012-2}{2} +1 = \frac{2010}{2} +1 = 1005+1 = 1006 \\ \\ \Longrightarrow~~n_1 = n_2 \\ \\ \text{Grupam termenii in grupe de cate 2 termeni, } \\ \text{din care, un termen pozitiv si un termen negativ.} [/tex]


[tex]S_4=3+5+7+9+\cdots + 2013 -2-4-6-8-\cdots - 2012 = \\ \\ =(3-2)+(5-4)+(7-6)+(9-8)+\cdots +(2013-2012) = \\ \text{(Avem 1006 grupe)} \\ \\ =\underbrace{1 + 1 + 1 + 1 + \cdots + 1 }_{\text{de 1006 ori}} = 1006 \cdot 1 = \boxed{1006}[/tex]



[tex]\displaystyle 4e) \\ S_5 = 1000+999-998-997+996+995-994-993+\cdots \\ \cdots +4+3-2-1 \\ \\ \text{De la 1000 la 1 sau de la 1 la 1000 sunt 1000 de termeni.} \\ \text{Grupam sirul in grupe de cate 4 termeni si obtinem:} \\ \\ n_g=\frac{1000}{4} = 250~\text{de grupe} \\ \\ S_5 = (1000+999-998-997)+(996+995-994-993)+\cdots \\ \cdots +(4+3-2-1) =\\ \\ \underbrace{4+4+ \cdots + 4}_{\text{De 250 de ori}} = 250 \cdot 4 = \boxed{1000} [/tex]