Question

Tính \(\left(\frac{1-\sqrt{5}}{2}\right)^{16}-\left(\frac{1+\sqrt{5}}{2}\right)^{16}\)

Answer 1

Đặt \(a=\frac{1-\sqrt{5}}{2},b=\frac{1+\sqrt{5}}{2}\)

Ta có \(a+b=1,a-b=-\sqrt{5},ab=-1\)

Ta sẽ tính từ từ. Cụ thể

\(a^2+b^2=\left(a+b\right)^2-2ab=3\)

\(a^2-b^2=\left(a+b\right)\left(a-b\right)=-\sqrt{5}\)

\(a^4+b^4=\left(a^2+b^2\right)^2-2\left(ab\right)^2=7\)

\(a^4-b^4=\left(a^2+b^2\right)\left(a^2-b^2\right)=-3\sqrt{5}\)

\(a^8+b^8=\left(a^4+b^4\right)^2-2\left(ab\right)^4=47\)

\(a^8-b^8=\left(a^4+b^4\right)\left(a^4-b^4\right)=-21\sqrt{5}\)

\(a^{16}-b^{16}=\left(a^8+b^8\right)\left(a^8-b^8\right)=-987\sqrt{5}\)