n*u_{2n+1} - u_{2n} + (n + 1)*u_{2(n+1)} = (n + 1)*u_{2(n+1)+1} - u_{2(n+1)}

Rewriting and rearanging these terms, we get:

n(u_{2n+1} + u_{2n+2}) + u_{2n+2} - u_{2n} = nu_{2n+3} + u_{2n+3} - u_{2n+2}

Now we need to try to break these up:

Recall that u_{k} + u_{k+1} = u_{k+2}, we will use this to rewrite many of the above Fibonacci Numbers.

n(u_{2n+1} + u_{2n+2}) +u_{2n+2}- u_{2n} = nu_{2n+3} +u_{2n+3}- u_{2n+2}

nu_{2n+3} +u_{2n+1} + u_{2n}- u_{2n} = nu_{2n+3} +u_{2n+1} + u_{2n+2}- u_{2n+2}

nu_{2n+3} + u_{2n+1} = nu_{2n+3} + u_{2n+1}

Q.E.D.