[tex]c) \\ -2-4-6- .....-2000= \\ =-2(1+2+3+.....+1000)= \\ =-2 \cdot \frac{1000 \; \cdot \;1001}{2} = -1000 \cdot 1001 = \boxed{-1001000}[/tex]
[tex]d) \\ 4563 \cdot 34 -4563 \cdot 200 +166 \cdot 4563 = \\ = 4563 (\underline{34}- 200+ \underline{166})=4563 (\underline{200}- 200)=4563 \cdot 0= \boxed{0}[/tex]
[tex]e) \\ 2\cdot (-1)^{k}+3\cdot (-1)^{k+1}-(-2)\cdot (-1)^{k+2}, \;\;\;\;\; k \in N \\ Precizari: \\ (-1) \; \text{ridicat la o putere impara = -1 } \\ (-1) \; \text{ridicat la o putere para = 1 } \\ \\ Avem \;2\;situatii: \\ Situatia\;1:\;\;\; (k) = par\;\;\;\; =\ \textgreater \ (k+1) = impar\;\;\;\; si \;\;\; (k+2) = par \\ =\ \textgreater \ \;\; 2\cdot (-1)^{k}+3\cdot (-1)^{k+1}-(-2)\cdot (-1)^{k+2}= \\ =2\cdot 1+3\cdot (-1)-(-2)\cdot 1=2-3+2 = 4-3= \boxed{1} [/tex]
[tex]Situatia\;2:\;\;\; (k) = impar\;\;\;\; =\ \textgreater \ (k+1) = par\;\;\;\; si \;\;\; (k+2) = impar \\ =\ \textgreater \ \;\; 2\cdot (-1)^{k}+3\cdot (-1)^{k+1}-(-2)\cdot (-1)^{k+2}= \\ =2\cdot (-1)+3\cdot 1-(-2)\cdot (-1)=-2+3-2 = -4 +3= \boxed{-1} [/tex]
