Question

Prove by mathematical induction x^n-y^n is divisible by x-y

Answer 1

Answer:

Step-by-step explanation:

Let P(n) denote the statement

$x^n-y^n$ is divisible by x-y

put n=1,

P(1): x - y is divisible by x-y

Assume the result is true for n=k

That is, $x^k-y^k$ is divisible by x-y

$\frac{x^k-y^k }{x-y}= c$

where c is an integer

$x^k-y^k=c(x-y)\\\\x^k=y^k+c(x-y)$

To prove P(k+1) is true

That is to prove:

$X^{k+1}-y^{k+1}$ is divisible by x-y

$X^{k+1}-y^{k+1}\\\\=x^k.x-y^{k+1}\\\\=[y^k+c(x-y)]x-y^{k+1}\\\\=xy^k+cx(x-y)-y^k.y\\\\=(x-y)y^k+cx(x-y)\\\\=(x-y)(y^k+cx)$, is divisible by x-y

Therefore P(k+1) is true

Hence by mathematical induction P(n) is true for all natural numbers.

Answer 2

Answer:

Step-by-step explanation:

Let P(n) denote the statement

is divisible by x-y

put n=1,

P(1): x - y is divisible by x-y

Assume the result is true for n=k

That is, is divisible by x-y

where c is an integer

To prove P(k+1) is true

That is to prove:

is divisible by x-y

, is divisible by x-y

Therefore P(k+1) is true

Hence by mathematical induction P(n) is true for all natural numbers