로그곡선 사다리꼴 넓이

[출제의도] $\displaystyle \sum\limits _{} ^{}$의 성질을 활용하여 문제해결하기 $\mathrm{A} ( \mathit{a} , 1 )$, $\mathrm{B} ( \mathit{b} , 0 )$이라 하면 $1 = \log _{2} a$, $0 = \log _{2} b$이므로 $\mathrm{A} ( \mathit{2} , 1 )$, $\mathrm{B} ( \mathit{1} , 0 )$ $\mathrm{P} _{\mathit{n}} ( \mathit{p} _{n} , 1 )$, $\mathrm{Q} _{\mathit{n}} ( \mathit{q} _{n} , 0 )$이라 하면 $\displaystyle 1 = \log _{2} \frac{p _{n}}{n}$, $\displaystyle 0 = \log _{2} \frac{q _{n}}{n}$이므로 $\mathrm{P} _{\mathit{n}} ( \mathit{2} n , 1 )$, $\mathrm{Q} _{\mathit{n}} ( \mathit{n} , 0 )$ 그러므로 $\displaystyle \mathrm{\overline{AP}} _{\mathit{n}} = \mathit{2} n - 2$, $\displaystyle \mathrm{\overline{BQ}} _{\mathit{n}} = \mathit{n} - 1$ $\displaystyle S _{n} = \frac{1}{2} \times \left( \mathrm{\overline{AP}} _{\mathit{n}} + \mathrm{\overline{BQ}} _{\mathit{n}} \right) \times 1$ $\displaystyle = \frac{1}{2} \times \left\{ \left( 2 n - 2 \right) + \left( n - 1 \right) \right\} \times 1 = \frac{3}{2} ( n - 1 )$ $\therefore$ $\displaystyle 2 \sum\limits _{n = 2} ^{10} S _{n}$$\displaystyle = 3 \sum\limits _{n = 2} ^{10} \left( n - 1 \right)$$\displaystyle = 3 \sum\limits _{n = 1} ^{9} n = 3 \times \frac{9 \times 10}{2} = 135$