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If \( A=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right] \), then use the principle of mathematical induction to show that
\[
A^{n}=\left[\begin{array}{ccc}
1 & n & n(n+1) / 2 \\
0 & 1 & n \\
0 & 0 & 1
\end{array}\right] \text { for every positive integer } n .
\]
\( \mathrm{P} \)
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