[tex]9)\\ \\ a)\quad \dfrac{2n-1+3\cdot (-1)^{n-1}}{4} \\ \\ \text{Daca n impar }\Rightarrow n = 2k+1,\quad k\in \mathbb{N}\\ \\ \Rightarrow \dfrac{2(2k+1)-1+3\cdot (-1)^{2k+1+1}}{4} = \dfrac{4k+2-1+3\cdot(-1)^{2(k+1)}}{4} = \\ \\ = \dfrac{4k+1+3}{4}=\dfrac{4k+4}{4}=k+1 \in \mathbb{Z} \\ \\ \text{Daca n par} \Rightarrow n = 2k,\quad k\in\mathbb{N}\\ \\ \Rightarrow \dfrac{2(2k)-1+3\cdot (-1)^{2k-1}}{4} = \dfrac{4k-1-3}{4} = \dfrac{4k-4}{4} = k-1 \in \mathbb{Z}[/tex]
=> Propozitia este adevarata.
[tex]b) \\ \\ |2^{33}-3^{22}|+|81^6-9^{11}|+|8^{11}-2^{32}|-3^{24}+2^{32} =\\\\ =|(2^{3})^{11}-(3^2)^{11}|+|(9^2)^6-9^{11}|+|(2^3)^{11}-2^{32}|-3^{24}+2^{32} =\\\\= |8^{11}-9^{11}|+|9^{12}-9^{11}| + |2^{33}-2^{32}|-3^{24}+2^{32}=\\\\=9^{11}-8^{11}+9^{12}-9^{11}+2^{33}-2^{32}-3^{24}+2^{32}=\\\\=(9^{11}-9^{11})+(2^{33}-8^{11})+(9^{12}-3^{24})+(2^{32}-2^{32})= \\ \\ = 0+0+0+0 = 0[/tex]
=> Propozitia este adevarata.
