Fun with implicit function and infinite sum

Algebra Level 3

\[\begin{cases} f:\mathbb{R}\rightarrow{}\mathbb{R} \\ f(0)=1 \\ f(xy+1)=f(x)f(y)-f(y)-x+2 \end{cases} \]

Evaluate

\[\sum_{i=1}^{\infty}{\frac{1}{2^{f(i-2)}}}\]

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