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NSWMaths Extension 1Quick questions

Proof (ME-P1)

Quick questions on Mathematical induction for divisibility: standard technique and algebraic restructuring

13short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the structure?
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To prove E(n)E(n) is divisible by dd for all positive integers nn:
What is the standard algebraic move?
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For E(k+1)E(k + 1), the most common technique is:
What is example pattern?
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For an−1a^n - 1 divisible by a−1a - 1 (for any integer aa): the standard manipulation is
What is different starting integer?
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If the statement is "for all n≥2n \ge 2" or "for all n≥5n \ge 5", start the base case at the smallest valid nn and adjust accordingly. The induction step still goes from kk to k+1k + 1.
What is 3n−13^n - 1 divisible by 22?
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Base (n=1n = 1): 3−1=23 - 1 = 2, divisible by 22.
What is n3+2nn^3 + 2 n divisible by 33?
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Base (n=1n = 1): 1+2=31 + 2 = 3, divisible by 33.
What is 4n+6n−14^n + 6 n - 1 divisible by 99?
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Base (n=1n = 1): 4+6−1=94 + 6 - 1 = 9, divisible by 99.
What is 22n−12^{2n} - 1 divisible by 33?
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Base (n=1n = 1): 4−1=34 - 1 = 3, divisible by 33.
What is divisible by a non-prime?
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Show 7n−3n7^n - 3^n divisible by 44 for all positive integers nn.
What is assuming what you want to prove?
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Do not write "assume 5k+1−1=4N5^{k + 1} - 1 = 4 N" as part of the step. The step derives this from the hypothesis on 5k−15^k - 1.
What is confusing the hypothesis with the conclusion?
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The hypothesis is about n=kn = k. The step derives the result at n=k+1n = k + 1.
What is missing the base case?
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Without a base case, the chain is unsupported.
What is algebra errors when expanding?
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(k+1)3=k3+3k2+3k+1(k + 1)^3 = k^3 + 3 k^2 + 3 k + 1, not k3+3k+1k^3 + 3 k + 1. Get the binomial expansion right. :::

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