A recursive sequence

Algebra Level 3

{a1=1an+1=2an+n2n,n1 \begin{cases} \begin{aligned}a_1&=1\\ a_{n+1}&=2a_{n}+n2^n, \quad n\ge 1 \end{aligned}\end{cases}

Let {ak} \left \{a_k\right \} be a sequence satisfying the above condition. Find the minimum nn such that an>2100a_n > 2^{100}.

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