SPS SPS FM 2024 October — Question 8 5 marks

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2024
SessionOctober
Marks5
TopicProof by induction
TypeProve divisibility
DifficultyStandard +0.3 This is a straightforward proof by induction with a divisibility statement. The base case is trivial (n=1 gives 77-13-1=63, divisible by 3), and the inductive step requires standard algebraic manipulation to factor out 3 from the difference. While it's a Further Maths question, it follows the standard induction template without requiring any novel insight or particularly complex algebra, making it slightly easier than average.
Spec4.01a Mathematical induction: construct proofs

Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by \(3\), for all integers \(n > 0\). [5]

Prove by induction that $11 \times 7^n - 13^n - 1$ is divisible by $3$, for all integers $n > 0$.
[5]

\hfill \mbox{\textit{SPS SPS FM 2024 Q8 [5]}}