Finding $n$th power of a $3\times 3$ matrix Find the $A^n$ if $$A=\begin{bmatrix}1 & a & b \\0 & 1 &a\\0 &0 &1\end{bmatrix}$$ 
I tried inductive method to show $$A^n=\begin{bmatrix}1 & na & nb+\frac{n(n-1)}{2}a^2 \\0 & 1 &na\\0 &0 &1\end{bmatrix}$$ 
now : My question is : Is there other method (idea ) to find $A^n$ ?
Thanks in advance.
Can the idea apply for $$A=\begin{bmatrix}1 & a & b \\0 & 1 &c\\0 &0 &1\end{bmatrix}$$ when $c \neq a$ ?

$$A=\begin{bmatrix}1 & a & b \\0 & 1 &c\\0 &0 &1\end{bmatrix}$$
Write $A = I + U$, where $U^2 = \begin{bmatrix} a \\ 0 \\ 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & c \end{bmatrix}$ and $U^3=0$.
$$ A^n=(I+U)^n=I+nU+{n\choose 2}U^2 =
\pmatrix{1&na&nb+\frac{n(n-1)}{2}ac\\0&1&nc\\0&0&1}$$
