등비수열과 로그의 합

$a _{n} = a \bullet r ^{n-1} ( a > 0 , r > 0 )$라 두면 $\displaystyle \sum\limits _{k = 1} ^{12} a _{k} = \frac{a ( r ^{12} - 1 )}{r - 1} = 100$ ……… ㉠ 수열 $\displaystyle \left\{ \frac{1}{a _{n}} \right\}$은 첫째항이 $\displaystyle \frac{1}{a}$이고, 공비가 $\displaystyle \frac{1}{r}$인 등비수열이므로 $\displaystyle \sum\limits _{k = 1} ^{12} \frac{1}{a _{k}} = \frac{\frac{1}{a} \left( 1 - \frac{1}{r ^{12}} \right)}{1 - \frac{1}{r}} = \frac{r ( r ^{12} - 1 )}{ar ^{12} ( r - 1 )} = 10$ …… ㉡ ㉠, ㉡을 연립하여 풀면, $a ^{2} r ^{11} = 10$ $\displaystyle \therefore \sum\limits _{k = 1} ^{12} \log a _{k} = \log ( a _{1} \times a _{2} \times \cdots \times a _{12} )$ $\displaystyle = \log ( a ^{12} r ^{\frac{11 \times 12}{2}} ) = \log ( a ^{12} r ^{66} )$ $= \log ( a ^{2} r ^{11} ) ^{6} = \log 10 ^{6} = 6$