Proof by induction question

Image and question below

On the 9th line: why not do 4 x 2^k = 8^k, then take out 8 from everything giving 8(6A - 1^k)?

Because the bold is wrong. Try it with k=2

Makes sense! Thanks

Must admit, it looks a bit laborious. You could write
2^(k+1) + 6^(k+1) = 2^(k+1)(1 + 3^(k+1))
as k>=1, the first term is 4*2^(k-1) and the (1+3^(k+1)) is even so overall divisible by 8. But not strictly induction.

Or a duffers proof, true for k=1 (8) and k=2 (40) and for k>=3, both 2^k and 6^k are divisible by 8, so so is their sum