ill skip a few steps...
Assume n=k
(3k+1)(7^k) -1 = 9M (where M is an integer) also 7^k = 9M+1/(3k+1)
for n=k+1
(3k+4)(7^k.7^1) - 1 sub in 7^k from above
(3k+4)(63M+1/3k+1) - 1
manipulate this and u get:
9(21kM + 2k + 28M +3) which is divisible by 9 because it is a common factor...