數學知識交流---求值 ( 計算方法1 )

(1) a # b = a/b + ab，求 3 # ( 12 # 4 ) 的值。
解：3 # ( 12 # 4 )
= 3 # ( 12/4 + 12×4 )
= 3 # ( 3 + 48 )
= 3 # 51
= 3/51 + 3×51
= 1/17 + 153
= 2602/17

(2) a $ b = a! - b! ，求 6 $ 5 - 5 $ 4 - 4 $ 3 - 3 $ 2 - 2 $ 1 的值。
解：6 $ 5 - 5 $ 4 - 4 $ 3 - 3 $ 2 - 2 $ 1
= (6! - 5!) - (5! - 4!) - (4! - 3!) - (3! - 2!) - (2! - 1!)
= 6! - 5! - 5! + 4! - 4! + 3! - 3! + 2! - 2! + 1!
= 6×5×4×3×2×1 - (5×4×3×2×1)×2 + 1
= 720 - 240 + 1
= 481

(3) a @ b = ( a + 1 ) / ( b - 1 )，x @ ( x @ 5 ) = 7/8 , 求 x @ x 的值。
解：x @ ( x @ 5 ) = 7/8
x @ [ (x + 1) / (5 - 1) ] = 7/8
x @ [ (x + 1) / 4 ]= 7/8
(x + 1) / [ (x + 1) / 4 - 1 ] = 7/8
(x + 1) / [ (x - 3) / 4 ] = 7/8
8 (x+1) = 7(x-3) / 4
32x + 32 = 7x - 21
25x = -53
x = -53/25
∴ x @ x
= (-53/25 + 1) / (-53/25 - 1)
= (-28/25) / (-78/25)
= 28/78
= 14/39