Skip to content

Proof by Induction Tutoring Session Prompt

Copy and paste this prompt to start your tutoring session about proof by induction:

Hi! I'm a student in CS 5001/5002 and I need help understanding proof by induction. We're currently in Week 4 of our integrated course where we're learning how mathematical induction connects to recursive programming and provides the theoretical foundation for understanding recursive algorithms.

My current understanding level:

  • I've watched the pre-class videos about proof by induction
  • I understand basic mathematical notation and algebra
  • I'm familiar with logical reasoning and if-then statements
  • I know what mathematical functions and sequences are
  • I understand the connection between induction and recursion (or want to learn it)

What I'd like to focus on today:

Please help me understand proof by induction by covering these topics in order:

  1. The Core Concept: What is mathematical induction? Why is it a valid proof technique? How is it like climbing an infinite ladder?

  2. The Structure: Every induction proof has two essential parts:

    • Base case: Prove the statement is true for the smallest value (usually n=1 or n=0)
    • Inductive step: Prove that if the statement is true for some value k, then it's also true for k+1

    Can you help me understand why these two parts together prove the statement for all values?

  3. The Inductive Hypothesis: What does it mean to "assume P(k) is true"? Why are we allowed to make this assumption?

  4. Classic Examples: Please walk me through these step-by-step:

    • Prove that 1 + 2 + 3 + ... + n = n(n+1)/2
    • Prove that 2^n > n for all n ≥ 1
    • Prove properties about Fibonacci numbers
    • Show me how each proof follows the two-step structure

  5. Connection to Recursion: How does the structure of an induction proof mirror the structure of a recursive function? Can you show me how:

    • Base case ↔ Base case in recursion
    • Inductive step ↔ Recursive case
    • The "leap of faith" in both approaches

  6. Common Mistakes: What are the typical errors students make, and how do I avoid them?

My specific questions:

[Add any specific questions you have here, such as:] - I don't understand why assuming P(k) doesn't make the proof circular - How do I know what to use as my base case? - I'm confused about the algebra in the inductive step - Can you help me prove [specific statement] by induction? - How does strong induction differ from regular induction? - Why can't I just use examples to prove something for all n?

Types of statements I want to practice:

  • Summation formulas (1+2+...+n, 1²+2²+...+n², etc.)
  • Inequality proofs (2^n > n, n! > 2^n for large n, etc.)
  • Divisibility statements (n³-n is divisible by 3, etc.)
  • Recursive sequence properties (Fibonacci, geometric series)
  • Set theory and combinatorics applications

Learning style preference:

  • I learn best with concrete numerical examples first
  • I prefer seeing the logical structure clearly laid out
  • I like connecting to programming/recursion analogies
  • I want to see common mistakes and how to avoid them
  • I need help with the algebraic manipulations

Please start with simple examples and build up gradually. I'd like to work through proofs step-by-step together, and I want to understand both the mechanical process and the deeper reasoning behind why induction works. Thanks!

Additional Resources

Pre-Class Videos (Review These First!)

Programming Connection

Quick Reference

Proof by Induction Template:

Statement to Prove: P(n) is true for all n ≥ n₀

Proof Structure: 

Proof by induction on n.

Base case: n = n₀
[Show that P(n₀) is true by direct verification]

Inductive step: 
Assume P(k) is true for some k ≥ n₀. (Inductive hypothesis)
We need to show that P(k+1) is true.
[Use the assumption P(k) to prove P(k+1)]

Conclusion: By mathematical induction, P(n) is true for all n ≥ n₀.

Common Induction Patterns:

  1. Summation Formula: 

    Base case: Verify the formula for n=1
Inductive step: 
  Assume: 1+2+...+k = k(k+1)/2
  Show: 1+2+...+k+(k+1) = (k+1)(k+2)/2

  LHS = [1+2+...+k] + (k+1)
      = k(k+1)/2 + (k+1)     [by inductive hypothesis]
      = (k+1)[k/2 + 1]
      = (k+1)(k+2)/2 = RHS ✓

  2. Inequality Proof: 

    Base case: Verify inequality for smallest n
Inductive step:
  Assume: f(k) > g(k)
  Show: f(k+1) > g(k+1)
  [Use properties of f and g, plus the assumption]

Induction vs Recursion Parallel

Mathematical Induction Recursive Programming
Base case: P(1) is true Base case: if n == 1: return base_value
Inductive step: P(k) → P(k+1) Recursive case: return function(n-1)
"Assume P(k), prove P(k+1)" "Trust that function(n-1) works"
Proves for all n ≥ 1 Computes for any valid input

Common Mistakes to Avoid

  • Circular reasoning: Don't use what you're trying to prove
  • Wrong base case: Make sure you start with the right value
  • Incomplete inductive step: Must use the inductive hypothesis
  • Algebraic errors: Be careful with manipulations
  • Assuming too much: Only assume P(k), not P(k+1)

Self-Check Questions

  • Did I clearly state what I'm proving?
  • Did I verify the base case by direct calculation?
  • Did I explicitly state my inductive hypothesis?
  • Did I use the inductive hypothesis in my proof of P(k+1)?
  • Are my algebraic steps correct and clearly explained?

This prompt is designed to help you get comprehensive help with proof by induction, connecting it to the recursive programming concepts we're learning in our integrated CS 5001/5002 course.