<s>[INST] Formula for $A^n$ where $n \in \{1, \ 2, \ \cdots \ \}$ for the matrix $A = \begin{bmatrix} 1 && b \\ 0 && 1 \end{bmatrix}$ Formula for $A^n$ where $n \in  \{1, \ 2, \ \cdots \ \}$ for the matrix $A = \begin{bmatrix} 1 && b \\ 0 && 1 \end{bmatrix}.$
Please help with the question if you can, it is for my Linear Algebra class and I cannot find anything close to an answer.

EDIT: As advised in comments, I have tried this for the first few $n$'s:
$A=\begin{bmatrix}
1 & b \\
0 & 1
\end{bmatrix}$
$A^2=
\begin{bmatrix}
1 & b \\
0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & b \\
0 & 1
\end{bmatrix}=
\begin{bmatrix}
1 & 2b \\
0 & 1
\end{bmatrix}$
$A^3=A^2\cdot A=
\begin{bmatrix}
1 & 2b \\
0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & b \\
0 & 1
\end{bmatrix}=
\begin{bmatrix}
1 & b+2b \\
0 & 1
\end{bmatrix}=
\begin{bmatrix}
1 & 3b \\
0 & 1
\end{bmatrix}$
 [/INST] $$A^n=\begin{pmatrix}1&nb\\0&1\end{pmatrix}$$
Suppose this is true until $n$. Then
$$A^{n+1}=A.A^n$$
Computing the right side almost completes the proof.
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