등비수열 곱

등비수열 $\left\{ a _{n} \right\}$의 첫째항을 $a$, 공비를 $r$ ($r > 0$)이라 하면 $2 a _{1} \left( a _{1} + a _{3} \right) = 20$에서 $2 a \left( a + ar ^{2} \right) = 20$ $\therefore$ $a ^{2} \left( 1 + r ^{2} \right) = 10$ $\cdots$$\cdots$ ㉠ 또한, $5 a _{2} \left( a _{1} + a _{2} \right) = 20$에서 $5 ar \left( a + ar \right) = 20$ $\therefore$ $a ^{2} r \left( 1 + r \right) = 4$ $\cdots$$\cdots$ ㉡ ㉠, ㉡에서 $\displaystyle \frac{a ^{2} \left( 1 + r ^{2} \right)}{a ^{2} r \left( 1 + r \right)} = \frac{10}{4}$, $\displaystyle \frac{1 + r ^{2}}{r \left( 1 + r \right)} = \frac{5}{2}$ $2 \left( 1 + r ^{2} \right) = 5 r \left( 1 + r \right)$, $3 r ^{2} + 5 r - 2 = 0$ $\left( 3 r - 1 \right) \left( r + 2 \right) = 0$ 이때 공비 $r$이 양수이므로 $\displaystyle r = \frac{1}{3}$ $\displaystyle r = \frac{1}{3}$을 ㉠에 대입하면 $\displaystyle a ^{2} \left( 1 + \frac{1}{9} \right) = 10$ $\therefore$ $a ^{2} = 9$ $\therefore$ $a _{1} \times a _{6}$$= a \times ar ^{5}$$= a ^{2} r ^{5}$$\displaystyle = 9 \times \left( \frac{1}{3} \right) ^{5}$$\displaystyle = \frac{1}{27}$