Prove the following by the principle of mathematical induction: `\ x^(2n-1)+y^(2n-1)` is divisible by `x+y` for all `n in Ndot`

Explanation

Let `P(n)=x^(2n)-y^(2n)`
Step I For `n=1`.
`P(1)=x^2-y^2=(x-y)(x+y)` which is divisible by `(x+y)`.
Therefore , the result is true for `n=1`.
Step II Assume that the result is true for `n=k`. Then ,
`P(k)=x^(2k)-y^(2k)` is divisible by `x+y`.
`rArr P(k)=(x+y)` r, where r is an integer.
Step III For `n=k+1`.
`=x^2.x^(2k)-y^(2).y^(2k)`
`=x^2x^(2k)-x^2y^(2k)+x^2y^(2k)-y^2y^(2k)`
`=x^(2)(x^(2k)-y^(2k))+y^(2k)(x^2-y^2)`
`=x^2(x+y)r+y^2k(x-y)(x+y)` [by assumption step]
`(x+y){x^2r+y^(2k)(x-y)}`
which is divisible by `(x+y) as x^2r+y^2k(x-y)` is an integer.
This shows that the result is true for `n=k+1`. Hence , by the principle of mathematical induction , the result is true for all `n in N`.