Mathematical Induction — Topic Review

Question 1 Fluency — Introduction to Induction

State the four steps of proof by mathematical induction.

Question 2 Fluency — Summation

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

Question 3 Fluency — Summation

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

Question 4 Understanding — Summation

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

Question 5 Fluency — Divisibility

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

Question 6 Understanding — Divisibility

Prove by mathematical induction that 9ⁿ − 1 is divisible by 8 for all integers n ≥ 1.

Question 7 Understanding — Divisibility

Prove by mathematical induction that 5ⁿ + 3 is divisible by 4 for all integers n ≥ 1.

Question 8 Problem Solving — Divisibility

Prove by mathematical induction that n³ + 2n is divisible by 3 for all integers n ≥ 1.

Question 9 Fluency — Inequalities

Prove by mathematical induction that 2ⁿ > n for all integers n ≥ 1.

Question 10 Understanding — Inequalities

Prove by mathematical induction that n! ≥ 2ⁿ⁻¹ for all integers n ≥ 1.

Question 11 Understanding — Inequalities

Prove by mathematical induction that 3ⁿ ≥ 2n+1 for all integers n ≥ 1.

Question 12 Problem Solving — Summation

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

Question 13 Understanding — Divisibility

Prove by mathematical induction that 2³ⁿ − 1 is divisible by 7 for all integers n ≥ 1.

Question 14 Problem Solving — Inequalities

Prove by mathematical induction that 2ⁿ ≥ n² for all integers n ≥ 4.

Question 15 Problem Solving — Summation

Prove by mathematical induction that ∑_k=1ⁿ k · k! = (n+1)! − 1 for all integers n ≥ 1.