공차가 분수인 등차수열

등차수열 $\left\{ a _{n} \right\}$의 첫째항을 $a$라 하자. $a _{n} = a + \left( n - 1 \right) d$ (단, $n$은 자연수) $a _{5} = a + 4 d$는 자연수이다. $\displaystyle S _{8} = \frac{8 \left( 2 a + 7 d \right)}{2} = 4 \left( 2 a + 7 d \right) = \frac{68}{3}$ $\displaystyle 2 a + 7 d = \frac{17}{3}$ $\displaystyle 2 \left( a + 4 d \right) - d = 2 a _{5} - d = \frac{17}{3}$ $\displaystyle a _{5} = \frac{1}{2} d + \frac{17}{6}$ $0 < d < 1$이므로 $\displaystyle \frac{17}{6} < a _{5} < \frac{10}{3}$ $a _{5} = 3$, $\displaystyle d = \frac{1}{3}$ $\displaystyle a _{5} = a + 4 \times \frac{1}{3} = 3$, $\displaystyle a = \frac{5}{3}$ $\displaystyle a _{n} = \frac{5}{3} + \left( n - 1 \right) \times \frac{1}{3}$ 따라서 $\displaystyle a _{16} = \frac{5}{3} + 15 \times \frac{1}{3} = \frac{20}{3}$