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in Mathematical Induction by (32.2k points)
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Prove that x2n – 1 + y2n – 1 is divisible by x + y for all n ϵ N.

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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.

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