Q. 383.7( 6 Votes )

Prove that x2n – 1 + y2n – 1 is divisible by x + y for all n ϵ N.

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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