SPS SPS FM Pure 2021 May — Question 5 5 marks

Exam BoardSPS
ModuleSPS FM Pure (SPS FM Pure)
Year2021
SessionMay
Marks5
TopicProof by induction
TypeProve divisibility
DifficultyModerate -0.3 This is a straightforward proof by induction with a simple divisibility statement. The base case is trivial, and the inductive step requires only basic algebraic manipulation (factoring out 7 and recognizing that 7≡3(mod 4) gives 7^k+3^{k-1}≡3·3^k+3^{k-1}≡4·3^{k-1}). While it requires understanding the induction framework, the arithmetic is routine and the problem follows a standard template, making it slightly easier than average.
Spec4.01a Mathematical induction: construct proofs

Prove by induction that, for all positive integers \(n\), \(7^n + 3^{n-1}\) is a multiple of \(4\). [5]

Prove by induction that, for all positive integers $n$, $7^n + 3^{n-1}$ is a multiple of $4$. [5]

\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q5 [5]}}