Solve S_{17}=1/2(1-(-2)^n)/1-(-2)

$$S _ { 17 } = \frac { \frac { 1 } { 2 } ( 1 - ( - 2 ) ^ { n } ) } { 1 - ( - 2 ) }$$

Solve for S_17 (complex solution)

$$S_{17}=\frac{1-\left(-2\right)^{n}}{6}$$

Solve for n (complex solution)

$$n=\frac{\ln(1-6S_{17})}{\ln(2)+\pi i}+\frac{2i\pi n_{1}}{\ln(2)+\pi i}$$

$$n_{1}\in \mathrm{Z}$$

$$S_{17}\neq \frac{1}{6}$$

Solve for S_17

$$S_{17}=\frac{1-\left(-2\right)^{n}}{6}$$

$$Denominator(n)\text{bmod}2=1$$

Solve for n

$$n=\log_{2}\left(1-6S_{17}\right)$$

$$S_{17}<\frac{1}{6}\text{ and }Numerator(\log_{2}\left(1-6S_{17}\right))\text{bmod}2=0\text{ and }Denominator(\log_{2}\left(1-6S_{17}\right))\text{bmod}2=1$$