등차수열 합 조건

$\displaystyle \sum\limits _{k = 1} ^{20} a _{2 k} - \sum\limits _{k = 1} ^{12} a _{2 k + 8}$ $= \left( a _{2} + a _{4} + \cdots + a _{40} \right) - \left( a _{10} + a _{12} + \cdots + a _{32} \right)$ $= a _{2} + a _{4} + a _{6} + a _{8} + a _{34} + a _{36} + a _{38} + a _{40}$ 등차중앙의 성질에 의해 등차수열 $\left\{ a _{n} \right\}$은 $m + l = 42$인 두 자연수 $m , l$에 대하여 $a _{m} + a _{l} = 2 a _{21}$을 만족시키므로 $\displaystyle \sum\limits _{k = 1} ^{20} a _{2 k} - \sum\limits _{k = 1} ^{12} a _{2 k + 8}$ $= \left( a _{2} + a _{40} \right) + \left( a _{4} + a _{38} \right) + \left( a _{6} + a _{36} \right) + \left( a _{8} + a _{34} \right)$ $= 2 a _{21} + 2 a _{21} + 2 a _{21} + 2 a _{21}$$= 48$ 그러므로 $a _{21} = 6$이고 $a _{3} + a _{39} = 2 a _{21} = 12$ 따라서 $a _{39} = 12 - a _{3} = 11$