Answer
All four sequences check the recurrence relation
Work Step by Step
We are given the recurrence relation:
$$\begin{align}a_n=-3a_{n-1}+4a_{n-2}.\end{align}\tag 1$$
We will check if the given sequences check that recurrence relation.
a) We substitute $a_{n-1}=0$ and $a_{n-2}=0$ in Eq. $(1)$ and we get:
$$-3(0)+4(0)=0\checkmark.$$
b) We substitute $a_{n-1}=1$ and $a_{n-2}=1$ in Eq. $(1)$ and we get:
$$-3(1)+4(1)=1\checkmark.$$
c) We substitute $a_{n-1}=(-4)^{n-1}$ and $a_{n-2}=(-4)^{n-2}$ in Eq. $(1)$ and we get:
$$\begin{align*}
-3(-4)^{n-1}+4(-4)^{n-2}&=-3(-4)^{n-1}-(-4)^{n-1}\\
&=-4(-4)^{n-1}\\
&=(-4)^n\checkmark.
\end{align*}$$
d) We substitute $a_{n-1}=2(-4)^{n-1}+3$ and $a_{n-2}=2(-4)^{n-2}+3$ in Eq. $(1)$ and we get:
$$\begin{align*}
-3[2(-4)^{n-1}+3]+4[2(-4)^{n-2}+3]&=-6(-4)^{n-1}-9+8(-4)^{n-2}+12\\
&=-6(-4)^{n-1}-2(-4)^{n-2}+3\\
&=-8(-4)^{n-1}+3\\
&=2(-4)^n+3\checkmark.
\end{align*}$$
