SPS SPS FM 2023 October — Question 8 5 marks

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2023
SessionOctober
Marks5
TopicProof by induction
TypeProve divisibility
DifficultyStandard +0.8 This is a proof by induction requiring students to manipulate algebraic expressions involving powers. While the technique is standard for Further Maths, the factorization step (extracting 8·2^{3k} - 3·3^k from 2^{3(k+1)} - 3^{k+1}) requires careful algebraic manipulation and insight to reveal the factor of 5, making it moderately challenging but within reach of well-prepared FM students.
Spec4.01a Mathematical induction: construct proofs

Prove that \(2^{3n} - 3^n\) is divisible by 5 for all integers \(n \geq 1\). [5]

Prove that $2^{3n} - 3^n$ is divisible by 5 for all integers $n \geq 1$. [5]

\hfill \mbox{\textit{SPS SPS FM 2023 Q8 [5]}}