Answer to Question #324189 in Discrete Mathematics for Mohammed AL braydi

Question

Given the recurrence relation 𝑎𝑛 =−2𝑎𝑛−1 +15𝑎𝑛−2 with the initial conditions 𝑎0=1 and 𝑎1=7.

(a) Write the characteristic equation.

(b) Solve the recurrence relation.

Answer 1

$a_n=-2a_{n-1}+15a_{n-2}\\a:\\\lambda ^2=-2\lambda +15\\\lambda ^2+2\lambda -15=0\\\lambda \in \left\{ -5,3 \right\} \\b:\\a_n=C_1\left( -5 \right) ^n+C_23^n\\\left\{ \begin{array}{c} a_0=1\\ a_1=7\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} C_1+C_2=1\\ -5C_1+3C_2=7\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} C_1=-0.5\\ C_2=1.5\\\end{array} \right. \\a_n=-0.5\left( -5 \right) ^n+1.5\cdot 3^n$