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VICSpecialist MathematicsQuick questions

Unit 3: Algebra, number and structure

Quick questions on Proof by mathematical induction: VCE Specialist Mathematics Unit 3

12short 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 are summation formulas?
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To prove ∑r=1nar=S(n)\sum_{r=1}^{n} a_r = S(n), the inductive step adds the (k+1)(k+1)th term to the assumed sum S(k)S(k) and simplifies to S(k+1)S(k+1).
What is divisibility?
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To prove that an expression is divisible by dd, write the assumption as "the expression at kk equals d×(integer)d \times (\text{integer})", then rearrange the k+1k+1 expression to expose a factor of dd.
What are inequalities?
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To prove f(n)≥g(n)f(n) \ge g(n), use the assumption f(k)≥g(k)f(k) \ge g(k) and bound the extra growth from kk to k+1k+1.
What is base step?
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The left side is ∑r=11r=1\sum_{r=1}^{1} r = 1. The right side is 1⋅22=1\frac{1 \cdot 2}{2} = 1. They agree, so P(1)P(1) is true.
What is inductive assumption?
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Assume the formula holds for n=kn = k, that is,
What is inductive step?
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Consider n=k+1n = k + 1. Split off the last term:
What is conclusion?
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The base step holds and the inductive step shows truth at kk implies truth at k+1k + 1, so by the principle of mathematical induction the formula holds for all integers n≥1n \ge 1.
What is never using the assumption?
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If the k+1k+1 working does not call on P(k)P(k), you have not done an induction proof. Point to where you substitute it.
What is no concluding statement?
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Finish with the sentence invoking the principle of mathematical induction, or you lose the final mark.
What is q1?
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State the three parts of a proof by mathematical induction. [2 marks]
What is q2?
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Verify the base step for ∑r=1nr2=n(n+1)(2n+1)6\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6} at n=1n = 1. [2 marks]
What is q3?
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In proving 5n−15^n - 1 is divisible by 44, write the inductive assumption. [1 mark]

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