\(u_{n+2}=7u_{n+1}-10u_n-4\)
\(\Leftrightarrow u_{n+2}-2u_{n+1}-1=5\left(u_{n+1}-2u_n-1\right)\)
Đặt \(u_{n+1}-2u_n-1=v_n\Rightarrow\left\{{}\begin{matrix}v_1=u_2-2u_1-1=1\\v_{n+1}=5v_n\end{matrix}\right.\)
\(\Rightarrow v_n\) là CSN với công bội \(q=5\)
\(\Rightarrow v_n=1.5^{n-1}=5^{n-1}\)
\(\Rightarrow u_{n+1}-2u_n-1=5^{n-1}\)
\(\Rightarrow u_{n+1}-\dfrac{1}{15}.5^{n+1}+1=2\left(u_n-\dfrac{1}{15}.5^n+1\right)\)
Đặt \(u_n-\dfrac{1}{15}.5^n+1=x_n\Rightarrow\left\{{}\begin{matrix}x_1=u_1-\dfrac{1}{15}.5+1=\dfrac{8}{3}\\x_{n+1}=2x_n\end{matrix}\right.\)
\(\Rightarrow x_n\) là CSN với công bội \(q=2\)
\(\Rightarrow x_n=\dfrac{8}{3}.2^{n-1}=\dfrac{4}{3}.2^n\)
\(\Rightarrow u_n=x_n+\dfrac{1}{15}.5^n-1=\dfrac{4}{3}.2^n+\dfrac{1}{15}.5^n-1\)
