Hint
The divisor is a nonzero integer. Use the difference-of-powers factorisation or exact integer arithmetic.
Worked solution
(a)
At \(\displaystyle n=1\) the difference is 4. Suppose \(7^k-3^k=4M\) for an integer M. Then \(7^{k+1}-3^{k+1}=7(7^k-3^k)+(7-3)3^k=4(7M+3^k)\) is divisible by 4. The successor and base case establish the result.
Answer: The difference is an integer multiple of 4 for every positive integer n.
(b)
\(7-3=4\)
Answer: \(4\).
Answer: \(4\)
(c)
\(\displaystyle \frac{7^{6}- 3^{6}}{4}= 29230\)
Answer: \(29230\).
Answer: \(29230\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 4 marks
- 1 mark: Verify the initial case on the specified integer domain.
- 1 mark: State and use the inductive assumption for an arbitrary k.
- 1 mark: Derive the required successor expression without assuming it.
- 1 mark: Conclude the claimed result for all integers in the stated domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A decimal approximation cannot prove exact divisibility.
(b) · 2 marks
- 1 mark: Evaluate the base case.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: State the base case and the integer domain.
(c) · 2 marks
- 1 mark: Use exact integer division after forming the power difference.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A decimal approximation cannot prove exact divisibility.
Total: 8 marks.