Standard +0.3 This is a straightforward proof by induction with a divisibility statement. The base case is trivial (n=1 gives 16+25+7=48, divisible by 3), and the inductive step requires standard algebraic manipulation to factor out 3 from f(k+1)-f(k). While it involves three terms with different bases, the technique is routine for Further Maths students who have practiced induction proofs.
Prove by induction that $f(n) = 2^{4n} + 5^{2n} + 7^n$ is divisible by 3 for all positive integers $n$.
[5]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q10 [5]}}