Induction and Binomials

🔮 Induction and Binomials: The Magic of Patterns

Imagine you’re a detective who discovers a secret pattern. Once you prove it works for one clue, you can prove it works FOREVER. That’s the superpower of mathematical induction!

🎯 What You’ll Master

• Base case
• Induction step
• Proving by induction
• Pascal’s triangle
• Binomial expansion
• General term formula
• Middle term
• Binomial coefficients

🧱 The Base Case: Your First Domino

Think of dominoes standing in a line. The base case is like tipping the FIRST domino.

What is a Base Case?

The base case is proving your statement works for the very first value (usually n = 1 or n = 0).

Example: Prove that 1 + 2 + 3 + … + n = n(n+1)/2

Base Case (n = 1):

• Left side: 1
• Right side: 1(1+1)/2 = 1(2)/2 = 1
• They match! ✅

It’s like checking if the first domino CAN fall. If it can, you’re ready for the next step!

🚀 The Induction Step: The Domino Effect

Here’s the magic: If one domino falls, it ALWAYS knocks down the next one.

How It Works

1. Assume your formula works for some number k (this is called the “induction hypothesis”)
2. Prove it MUST work for k+1

Example (continuing from above):

Assume it works for k: 1 + 2 + 3 + … + k = k(k+1)/2

Prove it works for k+1: 1 + 2 + … + k + (k+1) = ?

Take what we assumed and add (k+1): = k(k+1)/2 + (k+1) = k(k+1)/2 + 2(k+1)/2 = (k+1)(k+2)/2 = (k+1)((k+1)+1)/2 ✅

The formula works! If domino k falls, domino k+1 MUST fall too.

🎪 Proving by Induction: The Complete Show

graph TD
    A["🎯 Statement to Prove"] --> B["Step 1: Base Case"]
    B --> C{Does it work for n=1?}
    C -->|Yes ✅| D["Step 2: Induction Hypothesis"]
    D --> E["Assume works for k"]
    E --> F["Step 3: Induction Step"]
    F --> G["Prove works for k+1"]
    G --> H["🎉 PROVEN FOR ALL n!"]
    C -->|No ❌| I["Statement is False"]

The 3-Step Recipe

Example: Prove 2^n > n for all n ≥ 1

Base Case (n=1): 2^1 = 2 > 1 ✅

Assume for k: 2^k > k

Prove for k+1: 2^(k+1) = 2 × 2^k > 2k (using our assumption)

We need: 2k ≥ k+1 This means: k ≥ 1 ✅ (true for all k ≥ 1)

So 2^(k+1) > k+1 ✅ PROVEN!

🔺 Pascal’s Triangle: Nature’s Number Pattern

Imagine a magical mountain where each stone is the sum of the two stones directly above it!

          1
        1   1
      1   2   1
    1   3   3   1
  1   4   6   4   1
1   5  10  10   5   1

The Rules

1. Edges are always 1
2. Each inside number = sum of two numbers above
3. Row n has (n+1) numbers

Example: In row 4, the number 6 comes from 3 + 3 above it.

Hidden Treasures in Pascal’s Triangle

💥 Binomial Expansion: The Multiplication Shortcut

What if you need to calculate (a + b)^5 without multiplying 5 times?

The Pattern

(a + b)^n gives you a pattern with:

• n+1 terms
• Coefficients from Pascal’s triangle row n
• Powers of a decrease from n to 0
• Powers of b increase from 0 to n

Example: (a + b)^3

Row 3 of Pascal’s: 1, 3, 3, 1

(a + b)^3 = 1a³b⁰ + 3a²b¹ + 3a¹b² + 1a⁰b³ = a³ + 3a²b + 3ab² + b³

Quick Reference

🎯 General Term Formula: Find Any Term Instantly

Want to find the 5th term of (x + y)^10 without writing all terms? Use the general term formula!

The Formula

The (r+1)th term of (a + b)^n is:

T(r+1) = ⁿCᵣ × a^(n-r) × b^r

Where ⁿCᵣ = n! / (r! × (n-r)!)

Example: Find the 4th term of (x + 2)^6

Here: n = 6, r = 3 (because 4th term means r+1 = 4, so r = 3)

T₄ = ⁶C₃ × x^(6-3) × 2³ = 20 × x³ × 8 = 160x³

Step-by-Step Process

graph TD
    A["Which term? r+1"] --> B["Find r"]
    B --> C["Calculate ⁿCᵣ"]
    C --> D["Power of a = n-r"]
    D --> E["Power of b = r"]
    E --> F["Multiply all together!"]

⚖️ Middle Term: The Heart of the Expansion

Every expansion has a special center term (or two!).

Finding the Middle Term

Example 1: (x + y)^6 (n = 6, even)

• Middle term = (6/2 + 1)th = 4th term
• T₄ = ⁶C₃ × x³ × y³ = 20x³y³

Example 2: (a + b)^5 (n = 5, odd)

• Middle terms = 3rd and 4th terms
• T₃ = ⁵C₂ × a³ × b² = 10a³b²
• T₄ = ⁵C₃ × a² × b³ = 10a²b³

🔢 Binomial Coefficients: The Building Blocks

Binomial coefficients are the numbers in Pascal’s triangle. They tell you HOW MANY ways to choose items!

The Formula

ⁿCᵣ = n! / (r! × (n-r)!)

Also written as: C(n,r) or (n choose r)

Example: ⁵C₂ = 5! / (2! × 3!) = 120 / (2 × 6) = 10

Key Properties

Real-World Connection

“How many ways can you choose 2 toppings from 5 pizza toppings?”

Answer: ⁵C₂ = 10 ways! 🍕

🌟 Putting It All Together

graph TD
    A["Induction"] --> B["Prove patterns work FOREVER"]
    C[Pascal's Triangle] --> D["Shows binomial coefficients"]
    D --> E["Binomial Expansion"]
    E --> F["General Term Formula"]
    F --> G["Find ANY term quickly"]
    E --> H["Middle Term"]
    H --> I["Find the CENTER of expansion"]

Your New Superpowers

✅ Prove infinite patterns with just 2 steps ✅ Build Pascal’s triangle from memory ✅ Expand any (a + b)^n instantly ✅ Find any term without writing them all ✅ Calculate combinations for real problems

🎮 Quick Practice

1. What’s the base case for proving 1² + 2² + … + n² = n(n+1)(2n+1)/6?

2. What’s the 5th number in row 6 of Pascal’s triangle?

3. What’s the 3rd term in (x + 3)^4?

Answers: 1) n=1: 1² = 1 = 1(2)(3)/6 ✅ 2) 15 3) ⁴C₂ × x² × 9 = 54x²

You’ve just learned one of mathematics’ most powerful proof techniques AND the pattern behind polynomial expansion. These tools will serve you in algebra, calculus, probability, and beyond! 🚀