2 Related Questions About Finding A Closed Form

Question

1. Consider the sequence defined by $$ \begin{cases} s_0=0\\ s_1=3\\ s_n=6s_{n-1}-9s_{n-2} & \text{if }n\ge 2 \end{cases} .$$ Find a closed form for $s_n$.

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2. Consider the sequence defined by $$ \begin{cases} t_0=5\\ t_1=9\\ t_n=6t_{n-1}-9t_{n-2} & \text{if }n\ge 2 \end{cases} .$$ Find a closed form for $t_n$.

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I am having trouble with these questions about closed forms, could someone walk me step by step through each problem? Thanks!

Answer 1

for the first, begin by solving the characteristic equation $r^2-6r+9=0$

which has a double solution 3

so

$S_n=(an+b)3^n$

find a and b using initial conditions

$S_0$ and $S_1$.

Answer 2

Hint: For you first problem you could rewrite it as a matrix equation \begin{align} \begin{pmatrix} s_n\\ s_{n-1} \end{pmatrix} = \begin{pmatrix} 6s_{n-1}-9s_{n-2}\\ s_{n-1} \end{pmatrix} = \begin{pmatrix} 6 &-9\\ 1 & 0 \end{pmatrix} \begin{pmatrix} s_{n-1}\\ s_{n-2} \end{pmatrix} = \begin{pmatrix} 6 &-9\\ 1 & 0 \end{pmatrix}^{n-1} \begin{pmatrix} s_{1}\\ s_{0} \end{pmatrix}. \end{align} Thus, to find the closed form of $s_n$ it suffices to compute the $n-1$ power of your matrix.