Cho dãy số: \(U_n=\dfrac{\left(13+\sqrt{3}\right)^n-\left(13-\sqrt{3}\right)^n}{2\sqrt{3}}\)
b) CMR: \(U_{n+1}=26U_n-166U_{n-1}\)
P/s: Bài có 2 ý nhưng ý a mk bik lm rồi, còn ý b mn cố gắng giúp mk nha....
@Nguyễn Huy Tú, @Ace Legona, ngonhuminh,......
T lm nhé!
Ta có: \(U_{n+1}=\dfrac{\left(13+\sqrt{3}\right)^{n+1}-\left(13-\sqrt{3}\right)^{n+1}}{2\sqrt{3}}\)
\(=\dfrac{\left(13+\sqrt{3}\right)^n\cdot\left(13+\sqrt{3}\right)-\left(13-\sqrt{3}\right)^n\cdot\left(13-\sqrt{3}\right)}{2\sqrt{3}}\)
\(=\dfrac{\left(13+\sqrt{3}\right)^n\cdot\left(26+\sqrt{3}-13\right)-\left(13-\sqrt{3}\right)^n\left(26-\sqrt{3}-13\right)}{2\sqrt{3}}\)
\(=\dfrac{26\left(13+\sqrt{3}\right)^n+\sqrt{3}\left(13+\sqrt{3}\right)^n-13\left(13+\sqrt{3}\right)^n}{2\sqrt{3}}\)\(\dfrac{-26\left(13-\sqrt{3}\right)^n+\sqrt{3}\left(13-\sqrt{3}\right)^n+13\left(13-\sqrt{3}\right)^n}{.}\)
\(=\dfrac{26\left[\left(13+\sqrt{3}\right)^n-\left(13-\sqrt{3}\right)^n\right]}{2\sqrt{3}}\)\(+\dfrac{\sqrt{3}\left(13+\sqrt{3}\right)^n-13\left(13+\sqrt{3}\right)^n+\sqrt{3}\left(13-\sqrt{3}\right)^n+13\left(13-\sqrt{3}\right)^n}{2\sqrt{3}}\)
\(=\dfrac{26\left[\left(13+\sqrt{3}\right)^n-\left(13-\sqrt{3}\right)^n\right]}{2\sqrt{3}}+\dfrac{-\left[\left(13+\sqrt{3}\right)^n\left(13-\sqrt{3}\right)\right]}{2\sqrt{3}}+\dfrac{\left[\left(13-\sqrt{3}\right)^n\left(13+\sqrt{3}\right)\right]}{2\sqrt{3}}\)
\(=26U_n-\dfrac{166\left[\left(13+\sqrt{3}\right)^{n-1}-\left(13-\sqrt{3}\right)^{n-1}\right]}{2\sqrt{3}}\)
\(=26U_n-166U_{n-1}\) --> đpcm
P/s: Dấu = thứ 3 từ trên xuống cái p/s đấy là cả 1 dòng nha, tại dài quá nên ph chia lm 2 lần viết :v Lóa mắt
