(ii) Prove by induction that, for all positive integers n, f(n) = 3^(3n–2) + 2^(3n+1) is divisible by 19

Let P(n) represent the statement that 'f(n) is divisible by 19'. For the basis step, I prove that P(1) is true: f(1) = 3^3(1)-2+ 2³⁽¹⁾⁺¹ = 19. 19 is divisible by 19 so P(1) is true. I now want to prove that P(k) implies P(k+1) for all positive integers k. I therefore assume P(k), and I can write: 3^3k-2+ 2^3k+4 = 19m for some positive integer m. f(k+1) = 3^3k+1+2^3k+4 = 27(3^3k-2) + 8(3^3k+1) = 8(3^3k-2+2^3k+1) + 19(3^3k-2). I now substitute my assumption: f(k+1) = 19(8m + 3^3k-2). So P(k) implies P(k+1). Since P(1) is true, P(n) is therefore true for all positive integers n as required.