\(u_n-4u_{n-1}+3u_{n-2}=5.2^n\)
\(\Leftrightarrow u_n-u_{n-1}-3\left(u_{n-1}-u_{n-2}\right)=5.2^n\)
Đặt \(u_n-u_{n-1}=v_n\Rightarrow\left\{{}\begin{matrix}v_1=u_1-u_0=4\\v_n-3v_{n-1}=5.2^n\end{matrix}\right.\)
\(\Rightarrow v_n+10.2^n=3\left(v_{n-1}+10.2^{n-1}\right)\)
Đặt \(v_n+10.2^n=x_n\Rightarrow\left\{{}\begin{matrix}x_1=v_1+10.2^1=24\\x_n=3x_{n-1}\end{matrix}\right.\)
\(\Rightarrow x_n\) là CSN với công bội 3
\(\Rightarrow x_n=24.3^{n-1}\)
\(\Rightarrow v_n=x_n-10.2^n=24.3^{n-1}-10.2^n=8.3^n-10.2^n\)
\(\Rightarrow u_n-u_{n-1}=8.3^n-10.2^n\)
\(\Rightarrow u_n-12.3^n+20.2^n=u_{n-1}-12.3^{n-1}+20.2^{n-1}\)
Đặt \(u_n-12.3^n+20.2^n=y_n\Rightarrow\left\{{}\begin{matrix}y_1=u_1-12.3^1+20.2^1=7\\y_n=y_{n-1}=...=y_1=7\end{matrix}\right.\)
\(\Rightarrow u_n=12.3^n-20.2^n+7\)
