Question

find weather x^n+y^n is divisible by x-y (y not equal to 0) or not
​

Answer 1

Step-by-step explanation:

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

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.

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.

Read more on Brainly.in - https://brainly.in/question/6094785#readmore