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 is often used to prove formulas involving sums and algebraic identities.
Example:
Prove: 1 + 2 + 3 + ... + n = n(n + 1)/2 for all n ≥ 1.
Base Case (n = 1): LHS = 1, RHS = 1(1 + 1)/2 = 1 → true.
Induction Step: Assume the formula holds for n = k: 1 + 2 + ... + k = k(k + 1)/2.
Then for n = k + 1: 1 + 2 + ... + k + (k + 1) = [1 + 2 + ... + k] + (k + 1) = k(k + 1)/2 + (k + 1) (by induction hypothesis) = (k + 1)(k/2 + 1) = (k + 1)(k + 2)/2 = (k + 1)((k + 1) + 1)/2.
So the formula holds for k + 1.
This pattern—assume the identity for k, then add the next term—is standard in many summation proofs.
TL;DR — key idea
For algebraic identities, induction usually assumes the formula up to n = k and then adds the next term, simplifying to the desired expression for k + 1.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Use induction to prove ONE of the following algebraic identities: 1. 1 + 3 + 5 + ... + (2n − 1) = n². 2. 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6. Focus on writing a clear base case and a clean use of the induction hypothesis.
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.