로그함수 그래프와 넓이

[출제의도] 로그함수를 이용하여 추론하기 $\left| \log _{2} x - n \right| = 1$에서 $\log _{2} x - n = \mathit{1}$ 또는 $\log _{2} x - n = \mathit{-} 1$ $x = 2 ^{n + 1}$ 또는 $x = 2 ^{n - 1}$ $\left| \log _{2} x - n \right| = 2$에서 $\log _{2} x - n = \mathit{2}$ 또는 $\log _{2} x - n = \mathit{-} 2$ $x = 2 ^{n + 2}$ 또는 $x = 2 ^{n - 2}$ ㄱ. $\displaystyle {\overline{\mathrm{A} \mathit{\mathrm{_{1}}} \mathrm{B} \mathit{\mathrm{_{1}}}}} = 2 ^{2} - 2 ^{0} = 3$ (참) ㄴ. $\displaystyle {\overline{\mathrm{A} \mathit{\mathrm{_{n}}} \mathrm{B} \mathit{\mathrm{_{n}}}}} = 2 ^{n + 1} - 2 ^{n - 1} = 2 ^{n} ( 2 - 2 ^{- 1} )$$\displaystyle = \frac{3}{2} \times 2 ^{n}$ $\displaystyle {\overline{\mathrm{C} \mathit{\mathrm{_{n}}} \mathrm{D} \mathit{\mathrm{_{n}}}}} = 2 ^{n + 2} - 2 ^{n - 2} = 2 ^{n} ( 2 ^{2} - 2 ^{- 2} ) = \frac{15}{4} \times 2 ^{n}$ 따라서 $\displaystyle {\overline{\mathrm{A} \mathit{\mathrm{_{n}}} \mathrm{B} \mathit{\mathrm{_{n}}}}} : {\overline{\mathrm{C} \mathit{\mathrm{_{n}}} \mathrm{D} \mathit{\mathrm{_{n}}}}} = \frac{3}{2} : \frac{15}{4} = 2 : 5$ (참) ㄷ. $S _{n}$$\displaystyle = \frac{1}{2} \left( {\overline{\mathrm{A} \mathit{\mathrm{_{n}}} \mathrm{B} \mathit{\mathrm{_{n}}}}} + {\overline{\mathrm{C} \mathit{\mathrm{_{n}}} \mathrm{D} \mathit{\mathrm{_{n}}}}} \right) = \frac{1}{2} \left( {\overline{\mathrm{A} \mathit{\mathrm{_{n}}} \mathrm{B} \mathit{\mathrm{_{n}}}}} + \frac{5}{2} {\overline{\mathrm{A} \mathit{\mathrm{_{n}}} \mathrm{B} \mathit{\mathrm{_{n}}}}} \right)$ $\displaystyle = \frac{7}{4} {\overline{\mathrm{A} \mathit{\mathrm{_{n}}} \mathrm{B} \mathit{\mathrm{_{n}}}}} = \frac{21}{8} \times 2 ^{n}$ 따라서 $S _{n} = 21 \times 2 ^{n - 3}$ $21 \leq 21 \times 2 ^{k - 3} \leq 210$ $1 \leq 2 ^{k - 3} \leq 10$을 만족하는 자연수 $k$의 값은 $3$, $4$, $5$, $6$ 모든 자연수 $k$의 값의 합은 $18$ (거짓) 따라서 옳은 것은 ㄱ, ㄴ