mathbase
Learn/induction/The Logic Behind Induction

Lesson subsection

The Logic Behind Induction

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Mathematical induction is a proof technique used to show that a statement P(n) holds for all integers n greater than or equal to some starting value n₀.

The logic is like a line of dominoes:

  1. Base Case: Show P(n₀) is true. (Knock over the first domino.)
  2. Induction Step: Assume P(k) is true for an arbitrary k ≥ n₀ (induction hypothesis), and prove P(k + 1) is also true. (Each domino knocks over the next.)

If both parts hold, then:

  • P(n₀) is true,
  • P(n₀ + 1) is true,
  • P(n₀ + 2) is true,
  • and so on forever.

Induction does not prove infinitely many separate facts; instead, it proves a single logical pattern that propagates truth from one case to the next.

TL;DR — key idea

Induction proves a statement for all integers starting at n₀ by proving a base case and a step that pushes truth from n to n + 1.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

  • 1

    In your own words, explain the domino analogy for induction. Then describe what the base case and induction step represent in that analogy.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.