Mathematical Induction?
Prove by mathematical induction that
a - b is a factor of a^n - b^n.
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- Φ² = Φ+1Lv 73 weeks agoFavorite 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 sLv 73 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)
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