Task is to prove that at start of the N-round, count of hydras heads is divisible by 7. Rules of regrowing hydras heads are (all heads are cut off at start of each round, except black ones):
first head is brown (just 1 head), by cutting brown head will grow 3 black, 3 pink and 2 green heads, by cutting pink head will grow 3 black and 1 blue heads, by cutting green head will grow 1 black and 2 white heads, by cutting blue head will grow 3 blue heads, and by cutting white head will grow 4 white heads.
So every odd round from $n \geq 3$ is divisible by 7
My questin is: can i prove this divisiblity by base $n=14 + 3^{n-2} + 4^{n-2}$ and inductive step $n+2$ not $n+1$? or would you use another procces to prove this task?