CAIE Further Paper 1 2024 June — Question 2 6 marks

Exam BoardCAIE
ModuleFurther Paper 1 (Further Paper 1)
Year2024
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve divisibility
DifficultyStandard +0.8 This is a divisibility proof by induction requiring modular arithmetic with large numbers (6^{4n} and 38^n). The base case is straightforward, but the inductive step requires careful algebraic manipulation to factor out 74, which is more demanding than standard induction proofs. It's above average difficulty but not exceptional for Further Maths.
Spec4.01a Mathematical induction: construct proofs
2 Prove by mathematical induction that \(6 ^ { 4 n } + 38 ^ { n } - 2\) is divisible by 74 for all positive integers \(n\).
Question 2:
AnswerMarks Guidance
\(6^4 + 38 - 2 = 1332\) is divisible by 74B1 Checks base case
Assume that \(6^{4k} + 38^k - 2\) is divisible by 74 for some positive integer \(k\)B1 States inductive hypothesis
\(6^{4k+4} + 38^{k+1} - 2 = (1295+1) \times 6^{4k} + (37+1) \times 38^k - 2\)M1 A1 Separates \(6^{4k} + 38^k - 2\) or considers difference
Is divisible by 74 because \(1295 \times 6^{4k} + 37 \times 38^k\) is divisible by 74A1
Hence, by induction, \(6^{4k} + 38^k - 2\) is divisible by 74, is true for every positive integer \(n\)A1 
## Question 2:

| $6^4 + 38 - 2 = 1332$ is divisible by 74 | B1 | Checks base case |
| Assume that $6^{4k} + 38^k - 2$ is divisible by 74 for some positive integer $k$ | B1 | States inductive hypothesis |
| $6^{4k+4} + 38^{k+1} - 2 = (1295+1) \times 6^{4k} + (37+1) \times 38^k - 2$ | M1 A1 | Separates $6^{4k} + 38^k - 2$ or considers difference |
| Is divisible by 74 because $1295 \times 6^{4k} + 37 \times 38^k$ is divisible by 74 | A1 | |
| Hence, by induction, $6^{4k} + 38^k - 2$ is divisible by 74, is true for every positive integer $n$ | A1 | |

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2 Prove by mathematical induction that $6 ^ { 4 n } + 38 ^ { n } - 2$ is divisible by 74 for all positive integers $n$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q2 [6]}}
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