Let, P(n) be the given statement,
Now, P(n):x2n-1 + y2n – 1
Step1: P(1):x+y which is divisible by x+y
Thus,
P(1) is true.
Step2: Let, P(m) be true.
Then, x2m-1+y2m-1= λ(x+y)
Now, P(m+1) = x2m+1+y2m+1
= x2m+1+y2m+1-x2m-1.y2+x2m-1.y2
= x2m-1(x2-y2) + y2(x2m-1+y2m-1)
= (x+y)(x2m-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.