Mathematical Induction

[Jaspworld]

Prove that x^(2n+1) + y^(2n+1) is divisible by x+y for all integers n>(or equal to)0.

My solution
For 2k+1, we have
x^(2k+3)+y^(2k+3)
=x^2k*x^3+y^2k*y^3

=(x^2+y^2)(x^(2k+1)+y^(2k+1)) - (x^2)(y^(2k+1)) - (y^2)(x^(2k+1))
Now the first term is divisible by x+y but when I plug in values for k in the second and third terms they don't appear to be divisible by x+y. Where did I go wrong?

Thanks.

 

[pka]

This is what you need in the inductive step:
\(\displaystyle \L\begin{array}{rcl}
x^{2k + 3} + y^{2k + 3} & = & x^{2k + 3} + x^2 y^{2k + 1} - x^2 y^{2k + 1} + y^{2k + 3} \\
& = & x^2 \left( {x^{2k + 1} + y^{2k + 1} } \right) - y^{2k + 1} \left( {x^2 - y^2 } \right) \\
\end{array}\)