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Algebra Level 3

If the function \(f\) satisfies the relation \(f\left( x+y \right) = f\left( x \right) f\left( y \right) \) for all natural numbers \(x,y\)

And if \( f\left( 1 \right) =2 \), and for a natural number \(a\), the summation below is satisfied.

\( \large \displaystyle \sum _{ r=1 }^{ n }{ f\left( a+r \right) } =16\left( { 2 }^{ n }-1 \right) \)

What is the value of \(a\)?

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