Question

Tính \(\left(\frac{3+\sqrt{5}}{2}\right)^{^{1991}}+\left(\frac{3-\sqrt{5}}{2}\right)^{1991}-3\left(\frac{3+\sqrt{5}}{2}\right)^{1990}-3\left(\frac{3-\sqrt{5}}{2}\right)+\left(\frac{3+\sqrt{5}}{3}\right)^{1989}+\left(\frac{3-\sqrt{5}}{2}\right)^{1989}\)

Answer 1

Đặt \(\left\{{}\begin{matrix}a=\frac{3+\sqrt{5}}{2}\\b=\frac{3-\sqrt{5}}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}ab=1\\a+b=3\end{matrix}\right.\)

Đặt \(S_n=a^n+b^n\)

\(S_1=a+b=3\)

Ta cần tính \(S_{1991}-3S_{1990}+S_{1989}\)

Xét: \(S_1.S_n=\left(a+b\right)\left(a^n+b^n\right)=a^{n+1}+b^{n+1}+a.b^n+a^nb\)

\(\Rightarrow S_1S_n=a^{n+1}+b^{n+1}+ab\left(a^{n-1}+b^{n-1}\right)\)

\(\Leftrightarrow S_1S_n=a^{n+1}+b^{n+1}+a^{n-1}+b^{n-1}\)

\(\Leftrightarrow3S_n=S_{n+1}+S_{n-1}\)

Thay \(n=1990\Rightarrow3S_{1990}=S_{1991}+s_{1989}\)

\(\Rightarrow S_{1991}-3S_{1990}+S_{1989}=0\)