Prove by induction the following statements:

i) For every integer $n \geq 1$,

$$1^{2}+3^{2}+\cdots+(2 n-1)^{2}=\frac{1}{3}\left(4 n^{3}-n\right)$$

ii) For every integer $n \geq 1, n^{3}+5 n$ is divisible by 6 .