Proof by Mathematical Induction (Extension) — Topic Review

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This review covers all three lessons in the Proof by Mathematical Induction (Extension) topic: Induction for Series and Sums, Induction for Divisibility and Inequalities, and Induction for Matrices and Complex Numbers. Questions are exam-style and increase in difficulty.

Mixed Review Questions

1. Fluency

   Q1 — Sum of Integers

   Prove by mathematical induction that ∑_k=1ⁿ k = n(n+1)/2 for all positive integers n.

2. Fluency

   Q2 — Base Case for Divisibility

   Verify the base case for the statement: “3ⁿ − 1 is divisible by 2 for all n ≥ 1.”

3. Fluency

   Q3 — Inductive Hypothesis for Sum of Squares

   Write out the inductive hypothesis for proving ∑_k=1ⁿ k² = n(n+1)(2n+1)/6 by mathematical induction.

4. Fluency

   Q4 — De Moivre’s Theorem Application

   Evaluate (cos(π/4) + i sin(π/4))⁸ using De Moivre’s theorem.

5. Fluency

   Q5 — Sum of Odd Integers

   Prove by induction that ∑_k=1ⁿ (2k − 1) = n² for all positive integers n.

6. Understanding

   Q6 — Sum of Squares

   Prove by mathematical induction that ∑_k=1ⁿ k² = n(n+1)(2n+1)/6 for all positive integers n.

7. Understanding

   Q7 — Divisibility by 3

   Prove by induction that 4ⁿ − 1 is divisible by 3 for all n ≥ 1.

8. Understanding

   Q8 — Factorial Inequality

   Prove by induction that n! > 2ⁿ for all n ≥ 4.

9. Understanding

   Q9 — Power of a Matrix

   Prove by induction that ⎛1  2⎞ⁿ = ⎛1  2n⎞
   ⎝0  1⎠      ⎝0   1⎠
   for all positive integers n.

10. Understanding

    Q10 — De Moivre’s Theorem (Proof)

    Prove De Moivre’s theorem for positive integers: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for all n ∈ ℤ⁺.

11. Problem Solving

    Q11 — Power vs Square Inequality

    Prove by induction that 2ⁿ > n² for all n ≥ 5.

12. Problem Solving

    Q12 — Diagonal Matrix Power

    Prove by induction that ⎛3  0⎞ⁿ = ⎛3ⁿ   0 ⎞
    ⎝0  2⎠      ⎝0   2ⁿ⎠
    for all positive integers n.

13. Problem Solving

    Q13 — Triple Angle Formula

    Use De Moivre’s theorem with n = 3 to prove that cos(3θ) = 4cos³θ − 3cosθ.

14. Problem Solving

    Q14 — Sum Involving Powers of 2

    Prove by induction that ∑_k=1ⁿ k · 2^k = (n − 1) · 2ⁿ⁺¹ + 2 for all positive integers n.

15. Problem Solving

    Q15 — Telescoping Series

    Prove by induction that 1/(1·2) + 1/(2·3) + … + 1/(n(n+1)) = n/(n+1) for all positive integers n.