CAIE Further Paper 1 2022 November — Question 2 6 marks

Exam BoardCAIE
ModuleFurther Paper 1 (Further Paper 1)
Year2022
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve divisibility
DifficultyStandard +0.8 This is a divisibility proof by induction requiring students to handle two exponential terms with different bases (7^{2n} and 97^n) and show divisibility by 48. While the inductive step follows standard structure, manipulating the algebra to factor out 48 requires careful work with modular arithmetic or strategic factoring. It's harder than routine single-base divisibility proofs but remains a standard Further Maths induction question without requiring exceptional insight.
Spec4.01a Mathematical induction: construct proofs
2 Prove by mathematical induction that, for all positive integers \(n , 7 ^ { 2 n } + 97 ^ { n } - 50\) is divisible by 48. [6]
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
\(7^2 + 97 - 50 = 96\) is divisible by 48B1 Checks base case
Assume that \(7^{2k} + 97^k - 50\) is divisible by 48 for some positive integer \(k\)B1 States inductive hypothesis
Then \(7^{2k+2} + 97^{k+1} - 50 = (48+1)7^{2k} + (96+1)97^k - 50\)M1 A1 Separates \(7^{2k} + 97^k - 50\) or considers difference
is divisible by 48 because \(48 \times 7^{2k} + 96 \times 97^k\) is divisible by 48A1
Hence, by induction, true for every positive integer \(n\)A1 
## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $7^2 + 97 - 50 = 96$ is divisible by 48 | B1 | Checks base case |
| Assume that $7^{2k} + 97^k - 50$ is divisible by 48 for some positive integer $k$ | B1 | States inductive hypothesis |
| Then $7^{2k+2} + 97^{k+1} - 50 = (48+1)7^{2k} + (96+1)97^k - 50$ | M1 A1 | Separates $7^{2k} + 97^k - 50$ or considers difference |
| is divisible by 48 because $48 \times 7^{2k} + 96 \times 97^k$ is divisible by 48 | A1 | |
| Hence, by induction, true for every positive integer $n$ | A1 | |

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2 Prove by mathematical induction that, for all positive integers $n , 7 ^ { 2 n } + 97 ^ { n } - 50$ is divisible by 48. [6]\\

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q2 [6]}}
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