# 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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