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.
an=−2an−1+15an−2a:λ2=−2λ+15λ2+2λ−15=0λ∈{−5,3}b:an=C1(−5)n+C23n{a0=1a1=7⇒{C1+C2=1−5C1+3C2=7⇒{C1=−0.5C2=1.5an=−0.5(−5)n+1.5⋅3na_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^nan=−2an−1+15an−2a:λ2=−2λ+15λ2+2λ−15=0λ∈{−5,3}b:an=C1(−5)n+C23n{a0=1a1=7⇒{C1+C2=1−5C1+3C2=7⇒{C1=−0.5C2=1.5an=−0.5(−5)n+1.5⋅3n
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