등비수열과 절댓값 함수의 최솟값

$\displaystyle f ( x ) = \sum\limits _{n = 1} ^{17} \left| x - a _{n} \right|$은 $a _{i}$들의 중앙값인 $a _{9}$$= r ^{8}$에서 최소이므로 $r ^{8} = 16$, $\displaystyle r = \sqrt{2}$이다. 최솟값은 $\displaystyle m = \sum\limits _{n = 1} ^{8} ( 16 - a _{n} ) + \sum\limits _{n = 10} ^{17} ( a _{n} - 16 )$$\displaystyle = - \sum\limits _{n = 1} ^{8} a _{n} + \sum\limits _{n = 10} ^{17} a _{n}$ 인데, $\displaystyle \sum\limits _{n = 1} ^{8} a _{n} = \sum\limits _{n = 1} ^{8} r ^{n - 1}$$\displaystyle = \frac{r ^{8} - 1}{r - 1} = 15 ( \sqrt{2} + 1 )$ $\displaystyle \sum\limits _{n = 10} ^{17} a _{n} = r ^{9} \sum\limits _{n = 1} ^{8} a _{n} = 16 \sqrt{2} \times 15 ( \sqrt{2} + 1 )$ $\therefore$$r m$$\displaystyle \mathrm{=} \sqrt{2} \times 15 ( \sqrt{2} + 1 ) ( 16 \sqrt{2} - 1 )$$\displaystyle = 15 ( 30 + 31 \sqrt{2} )$