Master advanced mathematical induction techniques through comprehensive video lessons covering factorial identities, Fibonacci sequences, inequality proofs, derivative proofs, and geometric applications. Explore specialized induction methods including divisibility proofs, base cases above 1, increments greater than 1, and second-order recurrence formulas. Work through detailed examples proving inequalities such as Σ(k = 1 to n) 1/k² ≤ 2 - 1/n, 5^n + 9 < 6^n, and [2^(2n)]*(n!)^2 ≥ (2n)!, while learning to handle unusual properties of inequalities and different proof approaches. Study geometric applications including angle sum proofs for polygons and diagonal counting in convex polygons. Develop skills in proving divisibility statements like 8^n+2(7^n)-1 having a factor of 7 and a^n - b^n being divisible by a - b. Learn to work with recursive formulas, hypothesize explicit formulas, and apply induction to second-order recurrence relations. Practice exam-level problems and explore unconventional induction techniques including proofs by exhaustion and working with assumptions in complex inequality scenarios.

Syllabus

Induction: Factorial Identity
Induction: Fibonacci Sequence
Induction: Inequality Proofs
Induction: Derivative Proof
Induction: Geometry Proof (Angle Sum of a Polygon)
Induction: Diagonals in a Convex Polygon
Induction Inequality Proof Example 1: Σ(k = 1 to n) 1/k² ≤ 2 - 1/n
Induction Inequality Proof Example 2: n² ≥ n
Induction Inequality Proof Example 3: 5^n + 9 less than 6^n
Induction Inequality Proof Example 4: n! greater than n²
Induction Inequality Proof Example 5: 2^n ≥ n²
Induction Inequality Proof Example 6: [2^(2n)]*(n!)^2 ≥ (2n)!
Induction Geometry Proof: Diagonals in a Convex Polygon
Induction Inequality Proof Example 7: 4^n ≥ 1+3n
Induction: Divisibility Proof example 5 (8^n+2(7^n)-1 has a factor of 7)
Induction: Divisibility Proof example 6 (a^n - b^n is divisible by a - b)
Mathematical Induction (1 of 3: Outlining categories of induction, start of inequalities question)
Mathematical Induction (3 of 3: A different approach - rearranging the inequality)
Mathematical Induction (Harder Inequalities Proof by mathematical induction)
Mathematics Extension 1 Exam Review (2 of 3: Induction)
Induction Inequality Proofs (1 of 4: Unusual properties of inequalities)
Induction Inequality Proofs (2 of 4: Considering one side of the inequality)
Induction Inequality Proofs (3 of 4: Introducing & transforming the inequality)
Induction Inequality Proofs (4 of 4: Beginning with the assumption)
Mathematical Induction - Base case above 1 (1 of 3: Introduction)
Mathematical Induction - Base case above 1 (2 of 3: Completing the proof)
Mathematical Induction - Base case above 1 (3 of 3: Additional method)
Mathematical Induction (increments greater than 1)
Second Order Recurrence Formula (3 of 3: Proving k+2 case)
Second Order Recurrence Formula (2 of 3: An unusual test & assumption)
Second Order Recurrence Formula (1 of 3: Prologue - considering the old course)
Unusual Induction Inequality Proof (1 of 3: Base case)
Unusual Induction Inequality Proof (2 of 3: Working from the assumption)
Unusual Induction Inequality Proof (3 of 3: By exhaustion)
Recursive Formulas by Induction (2 of 4: Proving an explicit formula)
Recursive Formulas by Induction (1 of 4: What is a recursive formula?)
Recursive Formulas by Induction (4 of 4: Example HSC question)
Recursive Formulas by Induction (3 of 4: Hypothesising an explicit formula)