Fighter plane's analysis

Algebra Level 5

A fighter plane can travel \(d_n\) kilometers in the \(n\)-th hour. The values \(d_n\) are related by \(d_1 = 1\) and \(d_{n+1} = 2d_n + n(1+2^n)\) for all \(1 \le n \le 9\). If \(d_k = (k^2 - 2k + 13) \cdot 2^{k-2} - k - 1\), determine the value of \(k\).

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