I’ve looked at this question and the answer is:
an = (1/7)(6^n + 6.(-1)^n).a0 + (1/7)(6^n – (-1)^n).a1 + (7/8)(7^(n+1) + (-1)^n)) – 7*6^n
It is possible to do this problem manually, but the amount of work required is quite a lot.
I notice that you are doing a maths course as part of an engineering degree. So I suspect that maybe you are supposed/expected to use a computer algebra package such as Mathematica, Maple, Matlab, ....
This would solve your problem with much greater ease.