P(n) = 4n + 15n - 1. P(1) = 18. Divisible by 9
P(n+1) = P(n) + Q(n). (A) where Q(n) = 3*4n + 15
Q(n) = 3*4n + 15. Q(1) = 27. Divisible by 9
Q(n+1) = 3*4n+1 + 15. Q(n+1) = 12*4n + 15 Q(n+1) = 9*4n + Q(n).
Hence if Q(n) divisible by 9 then Q(n+1) divisible by 9
Hence Q(n) divisible by 9 (because Q(1) also divisible by 9)
So if P(n) divisible by 9 then P(n+1) divisible by 9 (because both terms on RHS of A are divisible by 9).
Since P(1) is also divisible by 9 then P(n) is divisible by 9. QED.