로그 곱 수열의 합

수열 $\left\{ a _{\mathit{n}} \right.$의 일반항 $a _{\mathit{n}} = 3 r ^{\mathit{n} - 1}$$\left( \mathit{r} > 1 \right)$ $b _{\mathit{n}}$$= \left( \log _{a _{1}} a _{2} \right) \times \left( \log _{a _{2}} a _{3} \right) \times \left( \log _{a _{3}} a _{4} \right) \times$$\cdots \times \left( \log _{a _{\mathit{n}}} a _{\mathit{n} + 1} \right)$ $\displaystyle = \frac{\log a _{2}}{\log a _{1}} \times \frac{\log a _{3}}{\log a _{2}} \times \frac{\log a _{4}}{\log a _{3}} \times \cdots \times \frac{\log a _{\mathit{n} + 1}}{\log a _{\mathit{n}}}$ $= \log _{a _{1}} a _{\mathit{n} + 1} = \log _{3} \left( 3 r ^{\mathit{n}} \right) = 1 + n \log _{3} r$ $\displaystyle \sum\limits _{\mathit{k} = 1} ^{10} b _{\mathit{k}}$$\displaystyle = 10 + \left( \log _{3} r \right) \times \sum\limits _{\mathit{k} = 1} ^{10} k$$= 10 + 55 \log _{3} r = 120$ 따라서 $\log _{3} r = 2$