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Core Pure

Quick questions on Proof by induction: sums, divisibility, recurrence and matrix powers - Edexcel A-Level Further Maths

6short 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 not using the assumption?
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The inductive step must use P(k)P(k) explicitly (substitute it in); a proof that re-derives P(k+1)P(k+1) from scratch is not a valid induction and scores no inductive-step marks.
What is vague divisibility?
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Show the result is a clear integer multiple of the divisor (write it as divisor times an integer), not just assert it is "divisible".
What is wrong base case?
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Start at the smallest value the statement claims, which is not always n=1n = 1; check the question for n≥0n \ge 0 or n≥2n \ge 2.
What is q1?
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Verify the base case for ∑r=1nr2=n(n+1)(2n+1)6\displaystyle\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}. [2 marks]
What is q2?
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State the assumption you make in the inductive step. [1 mark]
What is q3?
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In a divisibility proof, you reach f(k+1)=7f(k)+36f(k+1) = 7f(k) + 36 where the divisor is 66. Complete the argument. [2 marks]

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