سؤال #6,011

$$S_{n}= \frac{a_{1}(1 - q^n)}{1 - q} \Rightarrow \begin{cases} S_{8} = \frac{a_{1}(1 - q^{8} )}{1 - q} \\ S_{4} = \frac{a_{1}(1 - q^{4} )}{1 - q} \\ S_{8} = 10s_{4} \end{cases}\Rightarrow \frac{a_{1}(1 - q^{8} )}{1 - q} = \frac{10a_{1}(1 - q^{4} )}{1 - q} \Rightarrow $$ $$1- q^{8} = 10(1- q^{4} )\Rightarrow (1- q^{4} )(1+ q^{4} )= 10(1- q^{4} )\Rightarrow 1 + q^{4} = 10\Rightarrow q^{4} = 9\Rightarrow q^{8} = 81$$ $$\frac{a_{9}}{a_{1}}= \frac{a_{1}q^{8} }{a_{1}}= q^{8} = 81 \Rightarrow \frac{a_{9}}{a_{1}}= 81 \Rightarrow a_{9}= 81a_{1}$$
بنابراین گزینهٔ 3 پاسخ صحیح است.