등비수열 합 조건

등비수열 $\left\{ a _{n} \right\}$의 첫째항을 $a$, 공비를 $r$라 하자. $r = 1$이면 조건 (가)에서 $\displaystyle a = \frac{45}{4}$이고 조건 (나)에서는 $\displaystyle a = \frac{63}{2}$이므로 $r \ne 1$ $\displaystyle \sum\limits _{k = 1} ^{4} a _{k} = \frac{a ( r ^{4} - 1 )}{r - 1} = 45$ $\displaystyle \sum\limits _{k = 1} ^{6} \frac{a _{2} \times a _{5}}{a _{k}}$$\displaystyle = \left( a _{2} \times a _{5} \right) \times \sum\limits _{k = 1} ^{6} \frac{1}{a _{k}}$ $\displaystyle = ar \times ar ^{4} \times \frac{\frac{1}{a} \left\{ 1 - \left( \frac{1}{r} \right) ^{6} \right\}}{1 - \frac{1}{r}}$$\displaystyle = a ^{2} r ^{5} \times \frac{r ^{6} - 1}{a ( r ^{6} - r ^{5} )}$$\displaystyle = \frac{a ( r ^{6} - 1 )}{r - 1}$ $= 189$ $\displaystyle \frac{\frac{a ( r ^{6} - 1 )}{r - 1}}{\frac{a ( r ^{4} - 1 )}{r - 1}}$$\displaystyle = \frac{r ^{6} - 1}{r ^{4} - 1}$$\displaystyle = \frac{( r ^{2} - 1 ) ( r ^{4} + r ^{2} + 1 )}{( r ^{2} - 1 ) ( r ^{2} + 1 )}$$\displaystyle = \frac{r ^{4} + r ^{2} + 1}{r ^{2} + 1}$ $\displaystyle = \frac{189}{45}$ 이므로 $5 r ^{4} + 5 r ^{2} + 5 = 21 r ^{2} + 21$ $5 r ^{4} - 16 r ^{2} - 16 = 0$, $( 5 r ^{2} + 4 ) ( r ^{2} - 4 ) = 0$ $r > 0$이므로 $r = 2$ $\displaystyle \frac{a ( 2 ^{4} - 1 )}{2 - 1} = 15 a = 45$이므로 $a = 3$ 따라서 $a _{3} = 3 \times 2 ^{2} = 12$