How to Solve: Mathematical Induction & Inequalities (Wavy Curve, AM-GM, Cauchy-Schwarz) – IIT JEE Guide

How to Solve: Mathematical Induction & Inequalities (Wavy Curve, AM-GM, Cauchy-Schwarz) – IIT JEE Guide

Introduction

Mastering Mathematical Induction and Inequalities unlocks 10-15 marks in IIT JEE (Main + Advanced) – enough to push you from a 90th to a 99th percentile rank. These techniques solve problems where algebra alone fails, like proving nested inequalities, optimization, and divisibility in seconds.

What You Need To Know First

1. Basic Algebra – Factorization, quadratic equations, exponents.
2. Inequality Basics – Properties of <, >, ≤, ≥, and how to manipulate them.
3. Summation Notation – Understanding Σ (sigma) notation for series.

Key Vocabulary

Formulas To Know

1. Mathematical Induction

• Base Case: Prove P(1) is true.
• Inductive Hypothesis: Assume P(k) is true.
• Inductive Step: Prove P(k+1) is true using P(k).

MEMORISE THIS – The structure is always the same.

2. AM-GM Inequality

For non-negative real numbers a₁, a₂, ..., aₙ: (a₁ + a₂ + ... + aₙ)/n ≥ (a₁a₂...aₙ)^(1/n) Equality holds when all aᵢ are equal.

MEMORISE THIS – Most useful for 2 or 3 variables in JEE.

3. Cauchy-Schwarz Inequality

For real numbers a₁, a₂, ..., aₙ and b₁, b₂, ..., bₙ: (a₁b₁ + a₂b₂ + ... + aₙbₙ)² ≤ (a₁² + a₂² + ... + aₙ²)(b₁² + b₂² + ... + bₙ²) Equality holds when aᵢ = λbᵢ for some constant λ.

MEMORISE THIS – Often used in optimization problems.

4. Wavy Curve Method (Sign Analysis)

1. Find critical points (roots of the inequality).
2. Plot them on a number line.
3. Test intervals to determine where the inequality holds.

MEMORISE THIS – Works for polynomial and rational inequalities.

Step-by-Step Method

A. Mathematical Induction (Proving Statements for All n ∈ ℕ)

1. Write the statement P(n) clearly.
2. Example: P(n): 1 + 3 + 5 + ... + (2n-1) = n²
3. Prove the Base Case (n=1).
4. Substitute n=1 and verify.
5. Assume the Inductive Hypothesis (P(k) is true).
6. Write: "Assume P(k) holds: [statement for n=k]."
7. Prove P(k+1) using P(k).
8. Start with P(k+1), substitute P(k), and simplify.
9. Conclude by Induction.
10. Write: "By the principle of mathematical induction, P(n) is true for all n ∈ ℕ."

B. Wavy Curve Method (Solving Inequalities)

1. Rewrite the inequality in standard form (> 0, < 0, ≥ 0, ≤ 0).
2. Example: (x-1)(x-2) > 0
3. Find critical points (roots of the numerator and denominator).
4. For (x-1)(x-2) > 0, roots are x=1, x=2.
5. Plot critical points on a number line.
6. Draw a line with points at 1 and 2.
7. Test intervals between critical points.
8. Pick a test point in each interval (e.g., x=0, x=1.5, x=3).
9. Determine the sign of the expression in each interval.
10. Shade the regions where the inequality holds.
11. For > 0, shade where the expression is positive.
12. Write the solution in interval notation.
13. Example: x ∈ (-∞, 1) ∪ (2, ∞)

C. AM-GM Inequality (Proving Inequalities)

1. Identify non-negative variables.
2. Example: Prove a + 1/a ≥ 2 for a > 0.
3. Apply AM-GM to the terms.
4. (a + 1/a)/2 ≥ √(a × 1/a) = 1
5. Multiply both sides by 2.
6. a + 1/a ≥ 2
7. Check equality condition.
8. Equality holds when a = 1/a → a = 1.

D. Cauchy-Schwarz Inequality (Optimization Problems)

1. Identify two sequences (a₁, a₂, ..., aₙ) and (b₁, b₂, ..., bₙ).
2. Example: Maximize x + 2y given x² + y² = 1.
3. Write the Cauchy-Schwarz form.
4. (x×1 + y×2)² ≤ (x² + y²)(1² + 2²)
5. Substitute known values.
6. (x + 2y)² ≤ (1)(5) = 5
7. Take square roots.
8. x + 2y ≤ √5
9. Check equality condition.
10. Equality holds when x/1 = y/2 → y = 2x.
11. Substitute into x² + y² = 1 → x = ±1/√5, y = ±2/√5.

Worked Examples

Example 1 – Basic: Mathematical Induction

Problem: Prove 1 + 2 + 3 + ... + n = n(n+1)/2 for all n ∈ ℕ.

Solution: 1. Base Case (n=1):
- LHS = 1
- RHS = 1(2)/2 = 1
- LHS = RHS → True. 2. Inductive Hypothesis (n=k):
- Assume 1 + 2 + ... + k = k(k+1)/2. 3. Inductive Step (n=k+1):
- LHS = 1 + 2 + ... + k + (k+1)
- = k(k+1)/2 + (k+1) (using hypothesis)
- = (k+1)(k/2 + 1) = (k+1)(k+2)/2
- RHS = (k+1)(k+2)/2
- LHS = RHS → True. 4. Conclusion:
- By induction, the statement holds for all n ∈ ℕ.

What we did and why: - We followed the 3-step induction process (Base, Hypothesis, Step). - The key was substituting the hypothesis into the k+1 case.

Example 2 – Medium: Wavy Curve Method

Problem: Solve (x-1)(x+2)/(x-3) ≤ 0.

Solution: 1. Rewrite in standard form:
- (x-1)(x+2)/(x-3) ≤ 0 2. Find critical points:
- Numerator roots: x = 1, x = -2
- Denominator root: x = 3 (excluded, causes vertical asymptote) 3. Plot on number line:
- -∞ ... -2 ... 1 ... 3 ... ∞ 4. Test intervals:
- x = -3: (-4)(-1)/(-6) = -4/6 < 0 → Negative
- x = 0: (-1)(2)/(-3) = 2/3 > 0 → Positive
- x = 2: (1)(4)/(-1) = -4 < 0 → Negative
- x = 4: (3)(6)/(1) = 18 > 0 → Positive 5. Shade regions where ≤ 0:
- Includes -2 and 1 (since ≤ allows equality).
- Excludes 3 (denominator zero). 6. Solution:
- x ∈ [-2, 1] ∪ (3, ∞)

What we did and why: - We plotted critical points and tested intervals to determine where the inequality holds. - Denominator roots are always excluded (undefined).

Example 3 – Exam-Style: AM-GM + Cauchy-Schwarz

Problem: If a, b, c > 0 and a + b + c = 1, prove 1/a + 1/b + 1/c ≥ 9.

Solution: 1. Apply AM-HM (a variant of AM-GM):
- For positive numbers, (a + b + c)/3 ≥ 3/(1/a + 1/b + 1/c) 2. Substitute a + b + c = 1:
- 1/3 ≥ 3/(1/a + 1/b + 1/c) 3. Take reciprocals (reverse inequality since both sides positive):
- 3 ≤ (1/a + 1/b + 1/c)/3 4. Multiply by 3:
- 9 ≤ 1/a + 1/b + 1/c
- Thus, 1/a + 1/b + 1/c ≥ 9.

Equality condition: - Holds when a = b = c = 1/3.

What we did and why: - We used AM-HM (a direct consequence of AM-GM) to relate the sum and reciprocals. - The key was recognizing the constraint a + b + c = 1 to substitute.

Common Mistakes

Exam Traps

1-Minute Recap

Listen up! Tonight, before your exam, remember this:

1. Induction = 3 steps:
2. Prove n=1 (base case).
3. Assume n=k (hypothesis).
4. Prove n=k+1 (step).

5. Done! The statement is true for all n.

6. Wavy Curve = 4 steps:

7. Find roots (numerator and denominator).
8. Plot on a number line.
9. Test intervals.
10. Shade the solution.

11. Never include denominator roots!

12. AM-GM = 2 steps:

13. Write (a + b)/2 ≥ √(ab).
14. Check equality when a = b.

15. Works only for non-negative numbers!

16. Cauchy-Schwarz = 3 steps:

17. Write (a₁b₁ + ... + aₙbₙ)² ≤ (a₁² + ... + aₙ²)(b₁² + ... + bₙ²).
18. Substitute known values.
19. Check equality when aᵢ = λbᵢ.

Final tip: In JEE, inequalities often hide in optimization problems. If you see "maximum" or "minimum," think AM-GM or Cauchy-Schwarz first.

Now go crush that exam! ?