Question

PROVE:
1)
$x {}^{n} + y {}^{n} \: is \: divisible \: by \: x + y$
when n is odd.


BEST ANSWER WILL BE MARKED BRAINLIEST.
100 POINTS AWARDED FOR THE QUESTION.

Answer 1

Sorry dude I cannot do this by mathematical induction.

I have still so much to learn about mathematical induction.

Okay so:

Let f(x) be a function such that:

$f(x)=x^n+y^n$

If we put x as (-y) in the equation we will get:-

$\implies f(-y)=(-y)^n+y^n$

$\implies f(-y)=0$

We know that if f(x) is divisible by x-(-y) then f(-y)=0

So:

f(x) must be divisible by x+y

But we defined f(x) as $x^n+y^n$

So:

This proves that $x^n+y^n$ is divisible by x+y.

Another approach can be:

Putting n as 1 and 3 we get:

When n=1.

$x^n+y^n=x+y$

$\implies x^n+y^n=1(x+y)$

Since 1 is an integer it is divisible

Then if n=3

$x^n+y^n=x^3+y^3$

$\implies x^3+y^3=(x+y)(x^2-xy+y^2)$

Since $x^2-xy+y^2$ is an integer then it must be divisible.

So:

this proves your question.

Infact there is a formula if n is odd,

then:

$(x+y)[x^(n-1)-x^(n-2)y....................-xy^(n-2)+y^(n-1)]$

Remember that all xy multiples will be negative if x+y is the factor and in $x^n+y^n$ for every value of n as odd it will start by (x+y) multiplied by that formula.

For example:

$x^5+y^5\implies (x+y)(x^4-x^3y-x^2y^2-xy^3+y^4)$

Hope it helps:-)

Have a nice day and:

Buum kaali Happy Diwali:)

Answer 2

$<marquee>✌️REFER TO THE ATTACHMENT ✌️$