SPS SPS FM 2024 October — Question 8 6 marks

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2024
SessionOctober
Marks6
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 4+45=49), and the inductive step requires standard algebraic manipulation to factor out 7 from 2^{k+2} + 5×9^{k+1} using the inductive hypothesis. While it requires careful algebra, it follows a well-practiced template with no novel insight needed, making it slightly easier than average.
Spec4.01a Mathematical induction: construct proofs

Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]

Prove by induction that $2^{n+1} + 5 \times 9^n$ is divisible by 7 for all integers $n \geq 1$. [6]

\hfill \mbox{\textit{SPS SPS FM 2024 Q8 [6]}}