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MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

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12.5 Mathematical Induction

Verify a base case and prove an induction step.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Structure of a proof by inductionWorked steps · five questions02Induction for divisibility and inequalitiesWorked steps · five questions
Lesson brief and rules

12.5 Mathematical Induction

Prove a base case, then show P(k)P(k) implies P(k+1)P(k+1).

Worked example 11
01 / For P(n)P(n): 3(1+2+3(1+2+⋯+n)=3n(n+1)2+n)=3\dfrac{n(n+1)}{2}, check P(1)P(1).
02 / Left side is 33; right side is 3⋅1⋅22=33\cdot 1\cdot \dfrac{2}{2}=3.
03 / Answer: 3=33=3
Worked example 2 · Base case

For 1+1+⋯+n=n(n+1)2+n=\dfrac{n(n+1)}{2}, is n=1n=1 valid?

Left side is 11; right side is 1⋅22=11\cdot \dfrac{2}{2}=1. Answer: Yes

15 questions covering every idea, with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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