Induction Reference

Intro to Proofs · Axiom Academy

FORMULA SHEET Mathematical Induction Templates, formulas, and a troubleshooting guide for writing induction proofs. Inequality Induction Techniques Verify the base case carefully (it may need ). In the inductive step, combine P(k) with additional inequalities. Use algebraic manipulation to reach P(k+1) . Consider calculus (derivatives) to justify intermediate steps. Check whether the inequality is strict ( < ) or non-strict ( ). State what you're proving clearly. Identify the correct base case(s). Verify the base case(s) completely. State the inductive hypothesis precisely. Begin the inductive step with the P(k+1) expression. Actually use the inductive hypothesis. Show all algebraic steps clearly. Conclude by invoking the principle of induction. Check that quantifiers match (all n vs. some n ). Troubleshooting Guide: Common Errors Problem: forgot to verify P(1) , or checked the wrong starting value. Fix: always verify the smallest value in your domain. For inequalities the base might be n=2 , n=5 , etc. Problem: proved P(k+1) directly without assuming P(k) . Fix: you must explicitly use P(k) . If you didn't need it, induction isn't the right technique. Problem: assumed what you're trying to prove in the inductive step. Fix: start from the left side of P(k+1) , use P(k) , and manipulate to the right side. Don't assume both sides are equal. Problem: used weak induction when you need multiple previous cases.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.