Let, P(n) be the given statement,

Now, P(n):x^2n-1 + y^{2n – 1}

Step1: P(1):x+y which is divisible by x+y

Thus, P(1) is true.

Step2: Let, P(m) be true.

Then, x^2m-1+y^2m-1= λ(x+y)

Now, P(m+1) = x^2m+1+y^2m+1

= x^2m+1+y^2m+1-x^2m-1.y²+x^2m-1.y²

= x^2m-1(x²-y²) + y²(x^2m-1+y^2m-1)

= (x+y)(x^2m-1(x-y)+λy²)

Thus, P(m+1) is divisible by x+y. So, by the principle of mathematical

induction P(n) is true for all n.