110全國高中聯招

證明3
[解答]
\((1+x)^n=C_0^n+C_1^n x^1+C_2^n x^2+\ldots+C_n^n x^n\)
令\(x=i\)代入
\((1+i)^n=(C_0^n-C_2^n+C_4^n+\ldots)+(C_1^n-C_3^n+C_5^n-\ldots)i\)
則\(C_0^n-C_2^n+C_4^n+\ldots\)為\((1+i)^n\)的實部
　\(C_1^n-C_3^n+C_5^n-\ldots\)為\((1+i)^n\)的虛部
\(\displaystyle (1+i)^n=\left[\sqrt{2}\left(cos\frac{\pi}{4}+i sin\frac{\pi}{4}\right)\right]^n\)
\(=\sqrt{2}^n \left[cos \frac{n\pi}{4}+i sin \frac{n\pi}{4}\right]\)
\((1+i)^n\)的實部為\(\displaystyle \sqrt{2}^n cos \frac{n\pi}{4}\)，虛部為\(\displaystyle \sqrt{2}^n sin\frac{n\pi}{4}\)