Mathematical Induction?

Question

Prove by mathematical induction that

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

Answer 1

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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.

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Answer 2

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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Answer 3

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

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