My problems #7

Algebra Level 5

Define the function \(f(n)\) where \(n\) is a nonnegative integer satisfying \(f(0) = 1\) and \(f(n) \) is defined for \(n>0\) as \(f(n) = n\times{\displaystyle\sum_{i=0}^{n-1}\ f(i)}\).

Let \(2^k\) be the highest power of 2 that divides \(f(20)\). Find \(\sqrt{ 2^k}\).

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