세 항 곱의 무한급수

등비수열 $\left\{ a _{n} \right\}$의 첫째항을 $a$, 공비를 $r$라 하면 $\displaystyle a _{2} = ar = \frac{1}{2} \cdots \cdots \text{㉠}$ $\displaystyle a _{5} = ar ^{4} = \frac{1}{6} \cdots \cdots \text{㉡}$ ㉡÷㉠을 하면 $\displaystyle r ^{3} = \frac{1}{3}$ $\displaystyle \therefore r = ^{3} \sqrt{\frac{1}{3}} \cdots \cdots \text{㉢}$ ㉢을 ㉠에 대입하면 $\displaystyle a = \frac{{}^{3\sqrt{3}}}{2}$ $a _{n} a _{n+1} a _{n+2} = ar ^{n-1} \cdot ar ^{n} \cdot ar ^{n+1}$$\displaystyle = \frac{3}{8} \cdot \left( \frac{1}{3} \right) ^{n}$$= a ^{3} r ^{3n}$ $\displaystyle \sum\limits _{n=1} ^{\infty} a _{n} a _{n+1} a _{n+2} = \frac{\frac{1}{8}}{1 - \frac{1}{3}} = \frac{3}{16}$ ∴ $p + q = 16 + 3 = 19$