HSSLIVE Plus One Maths Chapter 4: Principle of Mathematical Induction Notes

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This chapter introduces students to a powerful proof technique that is essential for establishing mathematical truths about natural numbers. The principle of mathematical induction teaches students how to construct rigorous proofs through a two-step process: establishing a base case and then proving that if a statement is true for one case, it must be true for the next. This methodology cultivates logical reasoning, strengthens students’ ability to construct valid arguments, and introduces them to formal mathematical proof writing.

Chapter 4: Principle of Mathematical Induction

Mathematical induction is a method of mathematical proof used to establish that a statement is true for all natural numbers. It consists of two essential steps:

The Principle:

To prove that a statement P(n) is true for all natural numbers n ≥ n₀:

  1. Base Step: Prove P(n₀) is true.
  2. Inductive Step: Assume P(k) is true for some k ≥ n₀, then prove P(k+1) is also true.

If both steps are verified, then P(n) is true for all n ≥ n₀.

Applications:

Mathematical induction is commonly used to prove:

  1. Summation Formulas:
    • Sum of first n natural numbers: 1 + 2 + … + n = n(n+1)/2
    • Sum of squares: 1² + 2² + … + n² = n(n+1)(2n+1)/6
    • Sum of cubes: 1³ + 2³ + … + n³ = [n(n+1)/2]²
  2. Divisibility Properties:
    • n³ – n is divisible by 3 for all n ≥ 1
    • 2ⁿ > n for all n ≥ 1
  3. Inequalities:
    • (1 + x)ⁿ ≥ 1 + nx for x > -1 and n ≥ 1

Step-by-Step Approach:

  1. Clearly state the proposition P(n).
  2. Verify the base case (usually for n = 1 or another starting value).
  3. State the induction hypothesis (assume P(k) is true).
  4. Prove that P(k+1) is true using the induction hypothesis.
  5. Conclude that P(n) is true for all applicable values of n.

 

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