How can I use induction to show that all hexagonal numbers are odd? How can I use induction to show that all hexagonal numbers are odd? The following recurrence relation gives the hexagonal numbers: $1$ if $n=1$ and $H(n-1)+6n-6$ if $n>1$ I only want to use only the recurrence relation provided.

To prove this by induction, you need to show that

1. $H_1$ is odd, and
2. for all $n>1$, **if** $H_{n-1}$ is odd **then** $H_n$ is also odd.



Can you see how 2. follows from the recurrence relation (what can you say about $6n-6$)?