methods of characteristic equations

[Mahonroy]

Hello, I came across this problem and I wasn't entirely sure on how to do it. Or how I should start doing it anyways, so was seeing if anyone can point me in the right direction and give me some help. Any help is greatly appreciated, thanks!

 

[soroban]

Hello, Mahonroy!

\(\displaystyle \text{1, Apply the method of characteristic equation to find an explicit closed formula}\)

\(\displaystyle \text{for the numbers }a_0\text{ that satisfy the recurrence equation: }\;a_n \;=\;2a_{n-1} - a_{n-2}\;\text{ for }n \geq 2\)

. . \(\displaystyle \text{with the initial values: }\;a_0 = 4,\;a_1 = 1\)

Crank out the first few terms and a pattern appears . . .

. . \(\displaystyle \begin{array}{c|c} n & a_n \\ \hline 0 & 4 \\ 1 & 1 \\ 2 & \text{-}2 \\ 3 & \text{-}5 \\ 4 & \text{-}8 \\ 5 & \text{-}11 \\ 6 & \text{-}14 \\ \vdots & \vdots \end{array}\)

\(\displaystyle \text{The n}^{th}\text{ term is: }\;a_n \;=\;4-3n\)