등비수열 항 구하기

등비수열 $\left\{ a _{n} \right\}$의 공비를 $r$라 하면 $a _{1} a _{2} < 0$에서 $a _{1} \times \left( a _{1} r \right) = a _{1} ^{2} \times r < 0$ $\therefore$ $r < 0$ $a _{6} = 16$에서 $a _{1} r ^{5} = 16$ $\cdots$$\cdots$ ㉠ $2 a _{8} - 3 a _{7} = 32$에서 $2 a _{1} r ^{7} - 3 a _{1} r ^{6} = 32$, $a _{1} r ^{5} \left( 2 r ^{2} - 3 r \right) = 32$ $\cdots$$\cdots$ ㉡ ㉠, ㉡을 연립하면 $2 r ^{2} - 3 r = 2$, $\left( 2 r + 1 \right) \left( r - 2 \right) = 0$ $\therefore$ $\displaystyle r = - \frac{1}{2}$ $\left( \because r < 0 \right)$ $\therefore$ $a _{9} + a _{11}$$= a _{1} r ^{8} + a _{1} r ^{10}$$= a _{1} r ^{5} \left( r ^{3} + r ^{5} \right)$ $\displaystyle = 16 \times \left( - \frac{1}{8} - \frac{1}{32} \right)$ $\displaystyle = - \frac{5}{2}$