직사각형 수 배열 규칙

ㄱ. $< n , 1 > = n , < n , n > = 2 - n$ $< 10 , 1 > + < 10 , 10 > = 2$ (참) ㄴ. $\displaystyle < n , 2 > = \sum\limits _{k = 1} ^{n - 2} ( k + 1 )$ , $\displaystyle < n , n - 1 > = 2 + \sum\limits _{1} ^{n - 2} ( 1 - k )$ $\displaystyle < n , 2 > + < n , n - 1 > = 2 + \sum\limits _{k = 1} ^{n - 2} 2 = 2 n - 2$ $< 11 , 2 > + < 11 , 10 > = 20$(참) ㄷ. 제 $n$행의 수의 합 $S _{n}$을 $S _{n} = a _{1} + a _{2} + a _{3} + \cdots + a _{n}$라고 하면 $S _{n + 1} = ( a _{1} + 1 ) + ( a _{1} + a _{2} ) + ( a _{2} + a _{3} ) + \cdots + ( a _{n - 1} + a _{n} ) + ( a _{n} - 1 )$ $= 2 ( a _{1} + a _{2} + a _{3} + \cdots + a _{n} )$ $\therefore S _{n + 1} = 2 S _{n}$ , $S _{1} = 1$ $\therefore S _{n} = 2 ^{n - 1}$ ∴ $< 12 , 3 > + < 12 , 4 > + \cdots + < 12 , 10 >$ $= S _{12} - 2 - 22 = 2 ^{11} - 2 - 22 = 2024$ (참)