CAIE Further Paper 1 2020 November — Question 2 5 marks

Exam BoardCAIE
ModuleFurther Paper 1 (Further Paper 1)
Year2020
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve divisibility
DifficultyStandard +0.3 This is a straightforward divisibility proof by induction with a simple algebraic step. The base case is trivial (49-1=48), and the inductive step requires factoring 7^{2(k+1)}-1 = 49ยท7^{2k}-1 = 49(7^{2k}-1)+48, which directly shows divisibility by 12. While induction proofs require careful structure, this particular problem involves only routine algebraic manipulation without requiring deep insight or complex factorizations, making it slightly easier than average.
Spec4.01a Mathematical induction: construct proofs
2 Prove by mathematical induction that \(7 ^ { 2 n } - 1\) is divisible by 12 for every positive integer \(n\).
Question 2:
AnswerMarks Guidance
\(7^2 - 1 = 48\) is divisible by 12.B1 Checks base case.
Assume that \(7^{2k} - 1\) is divisible by 12 for some positive integer \(k\).B1 States inductive hypothesis.
Then \(7^{2k+2} - 1 = 49 \cdot 7^{2k} - 1 = 48 \cdot 7^{2k} + 7^{2k} - 1\) Or \((7^{2k+2} - 1) - (7^{2k} - 1) = 48 \cdot 7^{2k}\)M1 Separates \(7^{2k} - 1\).
\(7^{2k+2} - 1\) is divisible by 12.A1
So if true for \(n = k\) it is also true for \(n = k+1\). Hence, by induction, true for every positive integer \(n\).A1 Depends on previous M1 and A1.
Total: 5 
## Question 2:

| $7^2 - 1 = 48$ is divisible by 12. | B1 | Checks base case. |
|---|---|---|
| Assume that $7^{2k} - 1$ is divisible by 12 for **some** positive integer $k$. | B1 | States inductive hypothesis. |
| Then $7^{2k+2} - 1 = 49 \cdot 7^{2k} - 1 = 48 \cdot 7^{2k} + 7^{2k} - 1$ Or $(7^{2k+2} - 1) - (7^{2k} - 1) = 48 \cdot 7^{2k}$ | M1 | Separates $7^{2k} - 1$. |
| $7^{2k+2} - 1$ is divisible by 12. | A1 | |
| So if true for $n = k$ it is also true for $n = k+1$. Hence, by induction, true for every positive integer $n$. | A1 | Depends on previous M1 and A1. |
| **Total: 5** | | |

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2 Prove by mathematical induction that $7 ^ { 2 n } - 1$ is divisible by 12 for every positive integer $n$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q2 [5]}}
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Q1 8 Q2 5 Q3 7 Q4 13 Q5 14 Q6 13 Q7 15