Question

Să se calculeze suma: ex A9 -subpunctul c)
Clasa a 10-a , combinări

Answer 1

$c)\quad S_n = C_{n}^0+C_{n}^4+C_{n}^8+...\\ \\ (1+x)^n = C_{n}^0+C_{n}^1x+C_{n}^2x^2+C_n^3x^3+... \\ \\ \\(1+1)^n = C_{n}^0+C_{n}^1+C_{n}^2+C_n^3+C_n^4+...\\ \\ (1-1)^n = C_{n}^0-C_{n}^1+C_{n}^2-C_n^3+C_{n}^4-...\\ \\ (1+i)^n = C_{n}^0+C_{n}^1i-C_{n}^2-C_{n}^3i+C_{n}^4+C_{n}^5i-...\\ \\ (1-i)^n = C_{n}^0-C_{n}^1i-C_{n}^2+C_{n}^3i+C_{n}^4-C_{n}^5i-C_{n}^6+....\\ \\ \text{Adunam cele 4 relatii:} \\ \\ 2^n+0+(1+i)^n+(1-i)^n = 4(C_n^0+C_n^4+C_n^8+...)$

$\Rightarrow C_n^0+C_n^4+C_n^8+...= \dfrac{2^n+(1+i)^n+(1-i)^n}{4}\\ \\ \text{Folosind formula lui Moivre obtinem:}\\ \\ \boxed{C_{n}^0+C_{n}^4+C_{n}^8+... = \dfrac{2^n+2^{\frac{n+2}{2}}\cdot \cos\frac{n\pi}{4}}{4}}$