We will generalize Calvin Lin's answer a bit. Let
$$A_n = \begin{bmatrix} a & b & 0 & 0 & \cdots & 0\\ c & a & b & 0 & \cdots & 0\\ 0 & c & a & b & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & a & b\\ 0 & 0 & 0 & \cdots & c & a \end{bmatrix}.$$
We then have, by using Laplace expansion twice,
$$\det(A_n) = a \det(A_{n-1}) - bc \det(A_{n-2}).$$
Calling $\det(A_n) = d_n$ we have the following linear homogeneous recurrence relation:
$$d_n = a d_{n-1} - bc d_{n-2}.$$
The characteristic equation is
$$\begin{align}
x^2 - ax + bc = 0 & \implies \left(x - \frac{a}2 \right)^2 - \left(\frac{a}2 \right)^2 + bc = 0 \\
& \implies x = \frac{a \pm \sqrt{a^2-4bc}}2.
\end{align}$$
(This assumes a square roots exist. It's always the case in $\mathbb{C}$.)
Case 1: $a^2 - 4bc \neq 0$
In this case the characteristic polynomial has two distinct roots, so we have (for some constants $k_1$, $k_2$): $$d_n = k_1 \left( \dfrac{a + \sqrt{a^2-4bc}}2\right)^n + k_2 \left( \dfrac{a - \sqrt{a^2-4bc}}2\right)^n.$$
We have $d_1 = a$ and $d_2 = a^2 - bc$. We then get that $d_0 = 1$. Hence,
$$k_1 + k_2 = 1.$$
$$a (k_1 + k_2) + (k_1 - k_2)\sqrt{a^2-4bc} = 2a \implies k_1 - k_2 = \dfrac{a}{\sqrt{a^2-4bc}}.$$
Hence,
$$\begin{align}
k_1 & = \dfrac{a + \sqrt{a^2-4bc}}{2\sqrt{a^2-4bc}}, & k_2 & = -\dfrac{a-\sqrt{a^2-4bc}}{2\sqrt{a^2-4bc}}
\end{align}$$
And finally:
$$\color{red}{\det(A_n) = \dfrac1{\sqrt{a^2-4bc}} \left( \left( \dfrac{a + \sqrt{a^2-4bc}}2\right)^{n+1} - \left( \dfrac{a - \sqrt{a^2-4bc}}2\right)^{n+1}\right)}.$$
Plug in $a = 5$ and $b=c=2$ ($a^2 - 4 bc \neq 0$), to get
$$\det(A_n) = \frac{1}{3} ( 4^{n+1} - 1)$$
Case 2: $a^2 - 4bc = 0$
If the characteristic polynomial has a double root $x = a/2$, there exist constants $k_1$, $k_2$ such that:
$$d_n = (k_1 + k_2 n) \bigl(\frac{a}{2}\bigr)^n.$$
The initial conditions are $d_0 = 1$ and $d_1 = a$, thus:
$$\begin{align}
k_1 & = 1 & (k_1 + k_2) a = 2a
\end{align}$$
If $a = 0$, then $4bc = a^2$ implies either $b$ or $c$ is zero, and $d_n = 0$ for $n \ge 1$. Otherwise $$(k_1 + k_2) a = 2a \implies k_1 + k_2 = 2 \implies k_2 = 1.$$
And finally:
$$\color{red}{\det(A_n) = (n+1) \bigl(\frac{a}{2}\bigr)^n}.$$
We can simply calculate the determinant of an opposite (lower) triangular matrix:
Let $J_n$ be the $n \times n$ matrix with $1$ on the anti-diagonal and $0$ otherwise (i.e. $J_ne_i = e_{n+1-i}$ for all $1 \leq i \leq n$, where $e_1, \dotsc, e_n$ denotes the standard basis). Given any $m \times n$-matrix $A$ the matrix $AJ_n$ originates from $A$ by vertically mirroring its colums from the middle, i.e. swapping the first column with the last, the second with the second last, etc.
If $A$ is an $n \times n$ square matrix then we get from $J_n^2 = I_n$ that
$$
\det(A) = \det(J_n) \det(AJ_n).
$$
In the case of $D_n$ we get that $D_n J_n$ is the $a_n$-scalar multiple a lower triangular matrix with diagonal entries $x-a_1, x-a_2, \dotsc, x-a_{n-1}, 1$, so
$$
\det(D_n)
= \det(J_n) \det(D_n J_n)
= \det(J_n) a_n (x-a_1) \dotsm (x-a_{n-1}).
$$
So the only difference is that we need to know $\det(J_n)$. Because $J_n$ is a permutation matrix, corresponding to $\sigma_n \in S_n$ with $\sigma(i) = n+1-i$, we have $\det(J_n) = \mathrm{sgn}(\sigma_n)$. Notice that
\begin{align*}
\sigma_{2n} &= (1 \;\; 2n) (2 \;\; n-1) \dotsm (n \;\; n+1) \\
\sigma_{2n+1} &= (1 \;\; 2n+1) (2 \;\; n-1) \dotsm (n \;\; n+2).
\end{align*}
So we can just count the number of transpositions used and get that
$$
\mathrm{sgn}(\sigma_n) =
\begin{cases}
\phantom{-}1 & \text{if $n \equiv 0,1 \bmod 4$}, \\
-1 & \text{if $n \equiv 2,3 \bmod 4$},
\end{cases}
= (-1)^{n(n-1)/2}.
$$
So alltogether we have
$$
\det(D_n) = (-1)^{n(n-1)/2} a_n (x-a_1) \dotsm (x-a_{n-1}).
$$
(The nice thing about this is that now that we have calculated $\det(J_n) = (-1)^{n(n-1)/2}$ we can use this to calculate the determinant of opposite triangular and opposite diagonal matrices more ore less in the usual way.)
Best Answer
Beware : your proof isn't valid because you use implicitly a formula
$$\det(A+B)=\det(A)+\det(B)$$
which is erroneous. It's by chance that you get the right results.
Make instead a Laplace expansion of $D_n:=\det(M_n)$ with respect to its last column:
$$D_n=x\color{red}{(+1)}D_{n-1}+(-1)\color{red}{(-1)}\Delta \tag{1}$$
(the signs changing a minor into a cofactor are highlighted in $\color{red}{red}$) where $\Delta$ is a $(n-1) \times (n-1)$ determinant with last row $(1,0,\cdots 0)$. Expanding this determinant with respect to this row, you will get $\color{red}{(-1)^{n-2}}$ times a $(n-2) \times (n-2)$ matrix which is lower triangular, therefore whose determinant is the product of its diagonal elements, i.e., $(-1)^{n-2}$.
Consequently, (1) becomes :
$$D_n=xD_{n-1}+(-1)^{n-2}(-1)^{n-2} \ \ \iff \ \ D_n=xD_{n-1}+1\tag{2}$$
(with $D_1=x+1$). By a straightforward induction :
$$D_n=x(x\cdots (x(x+1)+1)\cdots)+1)+1\tag{3}$$
$$D_n=x^n+x^{n-1}+x^{n-2}+\cdots +x+1 \tag{4}$$
Remarks :
1) if you happen to know the concept of companion matrix, there is an easy alternative proof using the fact that $det(-M_n)$ is the characteristic polynomial of the companion matrix of the polynomial in the RHS of (4).
2) (3) can be called Horner factorization of (4).