Lecture Notes on Mathematical Induction Problem

Introduction

• Topic: Solving a problem using mathematical induction
• Equation: Sum of series up to n terms is equal to (16n^2(n+1)^2)
• Objective: Prove the equation is true using mathematical induction

Series Description

• The series given is of the form:
  • (S(n) = 4^3 + 8^3 + 12^3 + \ldots + (4n)^3)
• Each term is a cube of multiples of 4:
  • (4 \times 1^3, 4 \times 2^3, \ldots, 4 \times n^3)

Induction Proof

Step 1: Base Case

• Let (n = 1)
• LHS: (4 \times 1^3 = 4^3 = 64)
• RHS: (16 \times 1^2 \times (1+1)^2 = 16 \times 1 \times 4 = 64)
• Conclusion: LHS = RHS, hence true for (n = 1)

Step 2: Inductive Hypothesis

• Assume the statement is true for (n = k)
  • (S(k) = 4^3 + 8^3 + 12^3 + \ldots + (4k)^3 = 16k^2(k+1)^2)
• Conclusion: Assumption holds for (n = k)

Step 3: Inductive Step

• Show (S(n)) is true for (n = k+1)
• Find the next term: (4(k+1)^3)
• Add to the series:
  • (S(k+1) = S(k) + 4(k+1)^3)
  • Substitute:
    • (S(k+1) = 16k^2(k+1)^2 + 4(k+1)^3)
• Factorization and simplification:
  • Combine terms: (16(k+1)^2) appears as a common factor
  • Results in: (16(k+1)^2(k^2 + 4k + 4))
  • Simplifies to: (16(k+1)^2(k+1)^2)
• Conclusion: Proves true for (n = k+1)

Conclusion

• By the principle of mathematical induction, (S(n)) is true for all natural numbers (n)
• Completed for problem number four

Next Steps

• Next class will cover the solution to question number five
• Thanked viewers for watching the video