CAIE FP1 2019 June — Question 1 5 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2019
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve divisibility
DifficultyStandard +0.3 This is a straightforward divisibility proof by induction requiring students to show 3^(3(n+1)) - 1 - (3^(3n) - 1) is divisible by 13. The algebraic manipulation (factoring out 3^(3n) and showing 27-1=26 is divisible by 13) is routine for Further Maths students. While induction proofs require careful structure, this is a standard textbook exercise with no novel insight needed, making it slightly easier than average.
Spec4.01a Mathematical induction: construct proofs
1 Prove by mathematical induction that \(3 ^ { 3 n } - 1\) is divisible by 13 for every positive integer \(n\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(3^3 - 1 = 26\) is divisible by 13B1 Checks base case
Assume that \(3^{3k} - 1\) is divisible by 13 for some positive integer \(k\)B1 States inductive hypothesis
Then \(3^{3k+3} - 1 = 3^3 \cdot 3^{3k} - 1 = 26 \cdot 3^{3k} + 3^{3k} - 1\)M1 Separates \(3^{3k} - 1\)
is divisible by 13A1
\(H_k \Rightarrow H_{k+1}\). By induction, \(3^{3n} - 1\) is divisible by 13 for every positive integer \(n\)A1 States conclusion
Total: 5 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $3^3 - 1 = 26$ is divisible by 13 | **B1** | Checks base case |
| Assume that $3^{3k} - 1$ is divisible by 13 for some positive integer $k$ | **B1** | States inductive hypothesis |
| Then $3^{3k+3} - 1 = 3^3 \cdot 3^{3k} - 1 = 26 \cdot 3^{3k} + 3^{3k} - 1$ | **M1** | Separates $3^{3k} - 1$ |
| is divisible by 13 | **A1** | |
| $H_k \Rightarrow H_{k+1}$. By induction, $3^{3n} - 1$ is divisible by 13 for every positive integer $n$ | **A1** | States conclusion |
| **Total: 5** | | |

---
1 Prove by mathematical induction that $3 ^ { 3 n } - 1$ is divisible by 13 for every positive integer $n$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q1 [5]}}
This paper (12 questions)
View full paper
Q1 5 Q2 7 Q3 7 Q4 8 Q5 8 Q6 9 Q7 10 Q8 10 Q9 11 Q10 11 Q11 EITHER Q11 OR