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Algebra Level 4

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

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

r=1nf(a+r)=16(2n1) \large \displaystyle \sum _{ r=1 }^{ n }{ f\left( a+r \right) } =16\left( { 2 }^{ n }-1 \right)

What is the value of aa?

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