general expressions for halving

Where $n = 2^k, k \in \mathbb{N},$	$$\displaystyle\sum_{i=2}^{i=n} j x^{i} = {n-1 \over 2}$$	for $i j = n$
alternatively $f(0) = 0$,	$f(n+1) = 2f(n) + x^{{2}^{n+1}} - \frac{1}{2}$

Where \(n = 2^k, k \in \mathbb{N},\)	$$\displaystyle\sum_{i=2}^{i=n} j x^{i} = {n-1 \over 2}$$	for \(i j = n\)
alternatively \(f(0) = 0\),	\(f(n+1) = 2f(n) + x^{{2}^{n+1}} - \frac{1}{2}\)