A natural functional equation

Algebra Level 4

Find the natural number \(a\) for which \[\displaystyle \sum_{k=1}^n f(a+k)=16(2^n-1)\] where the function \(f\) satisfies the relation \(f(x+y)=f(x)f(y)\) for all natural numbers \(x,y\) and furthermore \(f(1)=2.\)

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