重酬!!! EASY MATHS (超趕gaga)

1)When n = 0 , 1 * 7^0 - 1 = 0 is divisible by 9.Assume that when n = k the statement is true i.e. :(3k+1)(7^k) - 1 = 9m , m is postive integer.When n = k+1 ,[3(k+1) + 1][7^(k+1)] - 1 = (3k+4) 7(7^k) - 1= (3k+4) 7(7^k) - [(3k+1)(7^k) - 9m]= (21k + 28 - 3k - 1)(7^k) + 9m= (18k + 27)(7^k) + 9m= 9 [(2k + 3)(7^k) + m] which is divisible by 9.By mathematical induction it is true for any integer. 2)When n = 0 , 7^0 - 1 = 0 is divisible by 2304.Assume that when n = k the statement is true i.e. :7^2k - 48k - 1 = 2304m , m is postive integer.When n = k+1 :7^[2(k+1)] - 48(k+1) - 1= 49[7^(2k)] - 48k - 49= 49(2304m + 48k + 1) - 48k - 49= 49(2304m) + 49*48k - 48k + 49 - 49= 49(2304m) + 48(49-1)k= 49(2304m) + 2304k= 2304(49m + k) which is divisible by 2304.By mathematical induction it is true for any integer.