Ratio

Let the number of terms, first term and common ratio of the GP be $n$ , $a_1 = a$ and $r$ respectively. Then, we have:

$\begin{aligned} \frac {S_{1 \to 11}}{S_{n-10 \to n}} & = \frac{a(r^{11}-1)}{r-1} \times \frac{r-1}{a_{n-10}(r^{11}-1)} = \frac{a}{a_{n-10}} = \frac{1}{8} \\ \Rightarrow \frac{a_{n-10}}{a} & = \frac{ar^{n-11}}{a} = \color{#3D99F6} {r^{n-11} = 8} \\ \frac {S_{10 \to n}}{S_{1 \to n-8}} & = \frac{a_{10}(r^{n-9}-1)}{r-1} \times \frac{r-1}{a(r^{n-9}-1)} = \frac{a_{10}}{a} = \color{#3D99F6} {r^9 = 2} \\ \color{#3D99F6} {r^{n-11}} & = \color{#3D99F6} {8 = 2^3 = \left( r^9\right)^3} \\ \Rightarrow n - 11 & = 27 \\ \Rightarrow n & = \boxed{38} \end{aligned}$