Finding $A^n$ for a matrix I have a matrix $$
A =
\left[ {\begin{array}{cc}
 1 & c  \\
 0 & d  \\
 \end{array} } \right]
$$
with $c$ and $d$ constant. I need to find $A^n$ ($n$ positive) and then need to prove that formula using induction.
I would like to check that the formula I derived is correct: 
$$
A^n =
\left[ {\begin{array}{cc}
 1 & c^{n-2}(dc + c)  \\
 0 & d^n  \\
 \end{array} } \right]
$$
If this is correct, how can I prove this? I suppose I can write $A^{n+1} = A^n A$, which would be
$$
\left[ {\begin{array}{cc}
 1 & c^{n-2}(dc + c)  \\
 0 & d^n  \\
 \end{array} } \right]
\left[ {\begin{array}{cc}
 1 & c  \\
 0 & d  \\
 \end{array} } \right]

$$
But then what would I do?
Thanks.

Letting $a_n$ be the upper right hand corner of $A^n$, and assuming it is obvious that the lower right corner of $A^n$ is $d^n$, and the left column is $[1,0]^T$, we get:
$$A^{n+1} = A^n A =
\left[ {\begin{array}{cc}
 1 & a_n  \\
 0 & d^n  \\
 \end{array} } \right]
\left[ {\begin{array}{cc}
 1 & c  \\
 0 & d  \\
 \end{array} } \right]$$
This gives us $a_{n+1} = c + d a_n$, with $a_1=c$.
In particular, then $a_n = c + cd + cd^2 + ... + cd^{n-1} = c\frac{d^n-1}{d-1}$.
So:
$$A^n =
\left[ {\begin{array}{cc}
 1 & c\frac{d^n-1}{d-1}  \\
 0 & d^n  \\
 \end{array} } \right]$$
