두 수열 귀납 관계

두 조건 (가)와 (나)로부터 $a _{2 n} + a _{2 n + 1} = 2 b _{n} + 1$ 두 조건 (다)와 (라)로부터 $b _{2 n} + b _{2 n + 1} = 2 a _{n} + 1$ $\displaystyle \sum\limits _{n = 1} ^{63} a _{n} - \sum\limits _{n = 1} ^{31} b _{n}$$\displaystyle = a _{1} + \sum\limits _{n = 1} ^{31} \left( a _{2 n} + a _{2 n + 1} \right) - \sum\limits _{n = 1} ^{31} b _{n}$ $\displaystyle = a _{1} + \sum\limits _{n = 1} ^{31} \left( 2 b _{n} + 1 \right) - \sum\limits _{n = 1} ^{31} b _{n}$ $\displaystyle = a _{1} + \sum\limits _{n = 1} ^{31} b _{n} + 31$ $\displaystyle = a _{1} + b _{1} + \sum\limits _{n = 1} ^{15} \left( b _{2 n} + b _{2 n + 1} \right) + 31$ $\displaystyle = a _{1} + b _{1} + \sum\limits _{n = 1} ^{15} \left( 2 a _{n} + 1 \right) + 31$ $\displaystyle = a _{1} + b _{1} + 2 \sum\limits _{n = 1} ^{15} a _{n} + 15 + 31$ $\displaystyle = a _{1} + b _{1} + 2 a _{1} + 2 \sum\limits _{n = 1} ^{7} \left( 2 b _{n} + 1 \right) + 46$ $\displaystyle = a _{1} + b _{1} + 2 a _{1} + 4 \sum\limits _{n = 1} ^{7} b _{n} + 14 + 46$ $\displaystyle = a _{1} + b _{1} + 2 a _{1} + 4 b _{1} + 4 \sum\limits _{n = 1} ^{3} \left( 2 a _{n} + 1 \right) + 60$ $\displaystyle = a _{1} + b _{1} + 2 a _{1} + 4 b _{1} + 8 \sum\limits _{n = 1} ^{3} a _{n} + 12 + 60$ $= a _{1} + b _{1} + 2 a _{1} + 4 b _{1} + 8 a _{1} + 8 ( 2 b _{1} + 1 ) + 72$ $= a _{1} + b _{1} + 2 a _{1} + 4 b _{1} + 8 a _{1} + 16 b _{1} + 8 + 72$ $= 11 a _{1} + 21 b _{1} + 80$$= 155$ 그러므로 $11 a _{1} + 21 b _{1} = 75$ $\cdots \cdots$ ㉠ 조건 (가)와 (다)에서 $a _{4 n} = b _{2 n} + 2 = ( 3 a _{n} - 2 ) + 2 = 3 a _{n}$ $b _{4 n} = 3 a _{2 n} - 2 = 3 ( b _{n} + 2 ) - 2 = 3 b _{n} + 4$ 이므로 $a _{48} = 3 a _{12} = 9 a _{3} = 9 ( b _{1} - 1 ) = 9$, $b _{1} = 2$ ㉠에 의하여 $a _{1} = 3$ 따라서 $b _{32} = 3 b _{8} + 4$$= 9 b _{2} + 16$$= 9 ( 3 a _{1} - 2 ) + 16$$= 79$