Lesson subsection
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.
Best flow: read → think on paper → write a short explanation → refine with feedback.
Induction can also prove inequalities, especially those involving growth rates.
Example:
Prove: 2ⁿ ≥ n + 1 for all integers n ≥ 0.
Base Case (n = 0): 2⁰ = 1 and 0 + 1 = 1 → true.
Induction Step: Assume 2ᵏ ≥ k + 1 for some k ≥ 0. Then: 2ᵏ⁺¹ = 2·2ᵏ ≥ 2(k + 1) = 2k + 2.
We want 2ᵏ⁺¹ ≥ (k + 1) + 1 = k + 2. But 2k + 2 ≥ k + 2 for k ≥ 0, so: 2ᵏ⁺¹ ≥ 2k + 2 ≥ k + 2.
Thus the inequality holds for k + 1.
Patterns for inequality induction:
TL;DR — key idea
Induction on inequalities uses the hypothesis to bound part of the expression for n + 1 and then shows that this bound is strong enough.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Use induction to prove ONE inequality: 1. 3ⁿ ≥ 2n + 1 for all n ≥ 0. 2. n! ≥ 2ⁿ for all n ≥ 4. Clearly show where you use the induction hypothesis in your argument.
Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.
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.