Mathematical Induction?

Prove by mathematical induction that

a - b is a factor of a^n - b^n.

3 Answers

Relevance
  • 3 weeks ago
    Best Answer

    a - b equals and so is a factor of a¹ - b¹.

    If a - b is a factor of aᵏ - bᵏ, so aᵏ - bᵏ = C(a - b), then

    aᵏ⁺¹ - bᵏ⁺¹ = aaᵏ - bbᵏ

    aᵏ⁺¹ - bᵏ⁺¹ = aaᵏ - abᵏ + abᵏ - bbᵏ

    aᵏ⁺¹ - bᵏ⁺¹ = a(aᵏ - bᵏ) + (a - b)bᵏ

    aᵏ⁺¹ - bᵏ⁺¹ = a(a - b)C + (a - b)bᵏ

    aᵏ⁺¹ - bᵏ⁺¹ = (a - b)(aC + bᵏ)

    So a - b is a factor of aᵏ⁺¹ - bᵏ⁺¹ if it is a factor of aᵏ - bᵏ,

    and a - b is a factor of a¹ - b¹,

    thus a - b is a factor of aⁿ - bⁿ for Integer n > 0.

  • ted s
    Lv 7
    3 weeks ago

    true if n = 1...assume true for n = k..ie . a^k - b^k = ( a - b ) W...then

    a^(k+1) - b^(k + 1 ) = a^(k+1) - a b^k + a b^k - b^(k+1) ≡a(a^k - b^k) + b^k ( a - b)

    = a (a-b)W +b^k ( a - b) = (a - b ) ( a W + b^k)

  • 3 weeks ago

    Prove by mathematical induction that a - b is a factor of a^n - b^n.

Still have questions? Get your answers by asking now.