등비수열과 곱 조건

조건 (가), (나)에 의해 상수 $a \left( a \ne 0 \right)$, $r \left( r > 0 \right)$에 대하여 $a _{1} = a$, $a _{3} = ar$, $a _{5} = ar ^{2}$, $a _{7} = ar ^{3}$ 조건 (나)에 의해 $\displaystyle a _{2} = \frac{75}{a _{7}} = \frac{75}{ar ^{3}}$, $\displaystyle a _{4} = \frac{75}{a _{5}} = \frac{75}{ar ^{2}}$ $\displaystyle a _{6} = \frac{75}{a _{3}} = \frac{75}{ar}$, $\displaystyle a _{8} = \frac{75}{a _{1}} = \frac{75}{a}$ $\displaystyle \sum\limits _{k = 1} ^{8} a _{k}$$= \left( a _{1} + a _{3} + a _{5} + a _{7} \right) + \left( a _{2} + a _{4} + a _{6} + a _{8} \right)$ $= \left( a + ar + ar ^{2} + ar ^{3} \right)$$\displaystyle + \left( \frac{75}{ar ^{3}} + \frac{75}{ar ^{2}} + \frac{75}{ar} + \frac{75}{a} \right)$ $\displaystyle = a \left( 1 + r + r ^{2} + r ^{3} \right) + \frac{75}{ar ^{3}} \left( 1 + r + r ^{2} + r ^{3} \right)$ $= \left( a _{1} + a _{2} \right) \left( 1 + r + r ^{2} + r ^{3} \right)$ $\displaystyle a _{1} + a _{2} = \frac{10}{3}$, $\displaystyle \sum\limits _{k = 1} ^{8} a _{k} = \frac{400}{3}$이므로 $\displaystyle \frac{10}{3} \left( 1 + r + r ^{2} + r ^{3} \right) = \frac{400}{3}$ $r ^{3} + r ^{2} + r - 39 = \left( r - 3 \right) \left( r ^{2} + 4 r + 13 \right) = 0$ $r$는 실수이므로 $r = 3$ $\displaystyle a _{1} + a _{2} = a + \frac{75}{ar ^{3}} = a + \frac{75}{27 a} = \frac{10}{3}$ $9 a ^{2} - 30 a + 25 = \left( 3 a - 5 \right) ^{2} = 0$, $\displaystyle a = \frac{5}{3}$ 따라서 $\displaystyle a _{3} + a _{8} = ar + \frac{75}{a} = 5 + 45 = 50$ [참고] $\displaystyle a _{1} = \frac{5}{3}$, $\displaystyle a _{2} = \frac{5}{3}$, $a _{3} = 5$, $a _{4} = 5$, $a _{5} = 15$, $a _{6} = 15$, $a _{7} = 45$, $a _{8} = 45$