Find the integer $n$ satisfying $\dfrac{{(n + 1)!\, - 3(n - 1)!}}{{(n + 1)!\, + 3(n - 1)!}} = \dfrac{{53}}{{59}}$ .
+128475 Cross-multiply
59 [ ( n + 1)! - 3(n - 1)! ] = 53 [ (n + 1)! +3 (n - 1)! ]
(59 - 53)(n + 1)! = (59*3 + 53*3) ( n -1)!
6(n + 1)! = 336 ( n -1)!
(n + 1)! / ( n -1)! = 336 / 6
(n + 1)! / ( n -1)! = 56 [ note : (n + 1)! / (n - 1)! = (n + 1) n ]
(n + 1) n = 56
8 * 7 = 56
n = 7
