Question

): Prove by mathematical induction. For any integer n ≥1, x3n – y3n is divisible by x - y, where x and y are any two integers with x ≠ y.

Answer 1

Answer:

do no the answer for your question

Answer 2

Given : . For any integer n ≥1,x³ⁿ - y³ⁿ  is divisible by x - y,

To Find : Prove by mathematical induction

Solution:

x³ⁿ - y³ⁿ  is divisible by x - y

n = 1

=> x³ - y³ = ( x - y) (x² + xy + y²)

Hence divisible by x - y

Assume that for n = a   a ∈ Z holds true  

x³ᵃ - y³ᵃ  is divisible by x - y

Hence x³ᵃ - y³ᵃ  = k(x - y)

=>  x³ᵃ = k(x - y) + y³ᵃ

n = a + 1

=> x⁽³ᵃ⁺³⁾ - y⁽³ᵃ⁺³⁾

= x³x³ᵃ - y³y³ᵃ

=>  x³ (k(x - y) + y³ᵃ) - y³y³ᵃ

= k x³.x  - k x³y  + x³ y³ᵃ - y³y³ᵃ

=  k x³(x - y)  + y³ᵃ(x³ -  y³)

=  k x³(x - y)  + y³ᵃ( x - y) (x² + xy + y²)

= (x - y) ( k x³ + y³ᵃ (x² + xy + y²))

Hence divisible by x - y

QED

Hence proved

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