Math: Discrete Mathematics Discrete Mathematics Question #2114 from sonam

Use induction to prove that x^(2n-1)+y^(2n-1) is divisible by x+y for n is the element of natural numbers

Expert's answer

For n =1:

$x^{2n-1}+y^{2n-1} = x+y$ - the condition is satisfied.
Let's assume that $x^{2n-1}+y^{2n-1}$
is divisible by (x+y), let's prove it for n+1:
$x^{2(n+1)-1}+y^{2(n+1)-1} = x^2x^{2n-1}+y^2y^{2n-1} = x^2x^{2n-1}+(y^2 + x^2 - x^2)y^{2n-1}=\\ =x^2(x^{2n-1}+y^{2n-1}) +(x+y)(x-y)y^{2n-1}$
The first term is divisible by assumption and it's shown that the second term is divisible by (x+y) too.

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Math: Discrete Mathematics Discrete Mathematics Question #2114 from sonam

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