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QLDSpecialist MathematicsQuick questions

Unit 3: Mathematical induction, and further vectors, matrices and complex numbers

Quick questions on Mathematical induction (QCE Specialist Mathematics Unit 3)

5short 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 principle?
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The principle of mathematical induction says: if P(1)P(1) is true (the base step), and if for every integer k≥1k \geq 1 the truth of P(k)P(k) implies the truth of P(k+1)P(k+1) (the inductive step), then P(n)P(n) is true for all positive integers nn.
What is base step?
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For n=1n=1: left-hand side =12=1= 1^2 = 1. Right-hand side =1⋅2⋅36=66=1= \dfrac{1 \cdot 2 \cdot 3}{6} = \dfrac{6}{6} = 1. So P(1)P(1) is true.
What is inductive hypothesis?
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Assume for some integer k≥1k \geq 1:
What is inductive step?
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Consider P(k+1)P(k+1). Add the next term (k+1)2(k+1)^2:
What is conclusion?
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Since P(1)P(1) is true and P(k)P(k) true implies P(k+1)P(k+1) true, by the principle of mathematical induction the result holds for all positive integers nn.

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