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Question

Prove that: x2ny2n is divisible by x+y.


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Solution

Let P(n):x2ny2n=(x+y)×d where dN

For n=1

LHS=x2×1y2×1=x2y2(x+y)(xy)=RHS

P(n) is true for n=1

Assume P(k) is true.

x2ky2k=(x+y)×mwheremN

We will prove that P(k+1) is true.

LHS=x2×(k+1)y2×(k+1)=x2k+2y2k+2=x2kx2y2ky2=(x+y)mx2+y2k(xy)=(x+y)×r

Where, r=mx2+y2k(xy)

P(k+1) is true whenever P(k) is true.

By the principle of mathematical induction, P(n) is true for n, where n is a natural number.


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