Question

Given the equation $h_n = 3h_(n-1)$, i obtained the characteristic equation $x-3=0$. How can I obtain the general solution?

Answer 1

\[
\begin{align*}
h_n&=3h_{n-1}\\
h_n-3h_{n-1}&=0\\
\lambda -3 &=0\\
\lambda &=3\\
\text{The general solution is } h_n=c\lambda^n=c3^n.\\
\end{align*}
\]
Suppose that a constant coefficient linear recurrence relations of degree n have a characteristic equation that have roots \(\lambda_i,\,1\leq i \leq n\) with no two roots equal (i.e. \(\lambda_i \neq \lambda_j,\,i\neq j\)), then the general solution to such relation is:\[
a_k=\sum_{r=1}^nc_r\lambda_r^k=c_1\lambda_1^k+c_2\lambda_2^k+c_3\lambda_3^k+\cdots +c_{n-1}\lambda_{n-1}^k+c_n\lambda_n^k
\]