If \( A=\left[\begin{array}{ll}a & b \\ 0 & 1\end{array}\right] \),...

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If \( A=\left[\begin{array}{ll}a & b \\ 0 & 1\end{array}\right] \), prove that \( A^{n}=\left[\begin{array}{cc}a^{n} & b\left(\frac{a^{n}-1}{a-1}\right) \\ 0 & 1\end{array}\right] \) for every positive integer \( n \).
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