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Question

x2n−1 + y2n−1 is divisible by x + y for all n ∈ N.

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Solution

Let P(n) be the given statement.
Now,
P(n): x2n-1+y2n-1 is divisible by x+y.Step1:P(1):x2-1+y2-1=x+y is divisible by x+yStep2:Let P(m) be true.Also,x2m-1+y2m-1 is divisible by x+y.Suppose: x2m-1+y2m-1=λx+y where λN ...(1)We shall show that Pm+1 is true whenever Pm is true.Now, Pm+1=x2m+1+y2m+1 =x2m+1+y2m+1-x2m-1.y2+x2m-1.y2 =x2m-1x2-y2+y2x2m-1+y2m-1 From (1) =x2m-1x2-y2+y2.λx+y =x+yx2m-1x-y+λy2 [It is divisible by (x+y).]Thus, Pm+1 is true.By the principle of mathematical induction, P(n) is true for all nN.

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