Challenging +1.2 This is a structured induction proof with the key step explicitly guided ('consider f(n+1) - f(n)'). Students must verify the base case, compute the difference algebraically using index laws, and show divisibility by 12. While it requires careful algebraic manipulation and understanding of induction structure, the heavy scaffolding and standard technique make it moderately above average but not particularly challenging for Further Maths students.
It is given that
$$\mathrm{f}(n) = 7^n (6n + 1) - 1.$$
By considering \(\mathrm{f}(n + 1) - \mathrm{f}(n)\), prove by induction that \(\mathrm{f}(n)\) is divisible by 12 for all positive integers \(n\). [6]
It is given that
$$\mathrm{f}(n) = 7^n (6n + 1) - 1.$$
By considering $\mathrm{f}(n + 1) - \mathrm{f}(n)$, prove by induction that $\mathrm{f}(n)$ is divisible by 12 for all positive integers $n$. [6]
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q2 [6]}}