[tex]S_n = C_{n}^0+C_{n}^4+C_{n}^8+...\\ \\ (1+x)^n = C_{n}^0+C_{n}^1x+C_{n}^2x^2+C_n^3x^3+... \\ \\ \\(1+1)^n = C_{n}^0+C_{n}^1+C_{n}^2+C_n^3+C_n^4+...\\ \\ (1-1)^n = C_{n}^0-C_{n}^1+C_{n}^2-C_n^3+C_{n}^4-...\\ \\ (1+i)^n = C_{n}^0+C_{n}^1i-C_{n}^2-C_{n}^3i+C_{n}^4+C_{n}^5i-...\\ \\ (1-i)^n = C_{n}^0-C_{n}^1i-C_{n}^2+C_{n}^3i+C_{n}^4-C_{n}^5i-C_{n}^6+....\\ \\ \text{Adunam cele 4 relatii:} \\ \\ 0+2^n+(1+i)^n+(1-i)^n = 4(C_n^0+C_n^4+C_n^8+...)[/tex]
[tex]\Rightarrow S_n= \dfrac{2^n+(1+i)^n+(1-i)^n}{4}\\ \\\Rightarrow S_n = \dfrac{2^n+\Big[\sqrt 2(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4})\Big]^n+\Big[\sqrt 2(\cos \frac{\pi}{4}-i\sin \frac{\pi}{4})\Big]^n}{4}\\ \\ \Rightarrow S_n = \dfrac{2^n+(\sqrt 2)^n(\cos \frac{n\pi}{4}+i\sin \frac{n\pi}{4})+(\sqrt 2)^n(\cos \frac{n\pi}{4}-i\sin \frac{n\pi}{4})}{4}[/tex]
[tex]\\ \\ \Rightarrow S_n = \dfrac{2^n+2\cdot 2^{\frac{n}{2}}\cdot \cos \frac{n\pi}{4}}{4} \\ \\ \Rightarrow S_n = \dfrac{1}{2}\Big(2^{n-1}+2^{\frac{n}{2}}\cos \frac{n\pi}{4}}\Big)[/tex]
=> e) corect
