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) " is divisible by " (x+y)`
for n=1 `P(1) : x^(2)=y^(2) =(x-y) (x+y)`
Which is divisible by x+y
`:.` P(n) is true for n=1
Let P (n) be trrue for n=1
`:. P (k) : x^(2k)-y^(2k)` ,is divisible by (x-y)
For n (K+1)
`P(K+1) : x^(2(K+1)) - y^(2(K+1))`
`=x^(2K+2)-y^(2K+2)`
`=x^(2k).x^(2)-y^(2k).y^(2)`
`=x^(2k).x^(2)-x^(2)y^(2k)+x^(2)y^2k)-y^(2k).y^(2)`
`=x^(2)(x^(2k)-y^(2k))+y^2k)(x^(2)-y^(2))`
`:. P(k)` is true
`:. x^(2)(x^(2k)-y^(2k))` is divisible by (x+y) and `y^(2k)(x-y)`
`(x+y)` is also divisible by `(x+y)`.
`:.x^(2)(x^(2k)-y^(2k))+y^(2k)(x-y)(x-y)` is divisible by (x+y)
`rArr` P (n) is also true for n=K+1
Hence from the principle of mathematical induction P (n) is true for all vlaues of n where `n in N`