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 3. Suppose \(8^k-5^k=3M\) for an integer M. Then \(8^{k+1}-5^{k+1}=8(8^k-5^k)+(8-5)5^k=3(8M+5^k)\) is divisible by 3. The successor and base case establish the result.
Answer: The difference is an integer multiple of 3 for every positive integer n.
(b)
\(8-5=3\)
Answer: \(3\).
Answer: \(3\)
(c)
\(\displaystyle \frac{8^{5}- 5^{5}}{3}= 9881\)
Answer: \(9881\).
Answer: \(9881\)
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.