How do I write $\{3,6,11,18,27,38,…\}$ in set-builder notation?

Question

For the set $\{3,6,11,18,27,38,...\}$, the $(n+1)^\text{th}$ term, which I'll call $a_{n+1}$, is: $$a_{n+1} = a_n + 2n+1$$ How can I write this set in set-builder notation? My best guess doesn't seem quite right: $$\{a_n : n \in \mathbb{N}, a_1=3, a_{n+1}=a_n + 2n+1\}$$ Also, my reference to $\mathbb{N}$ assumes $0 \notin \mathbb{N}$, but maybe that's the less common interpretation of $\mathbb{N}$?

Thanks for the help! As pointed out below, $\{n^2+2:n\in \mathbb{N}\}$ does the trick nicely. Bit of an oversight on my part :P

Answer 1

How about $$\{ n^2 + 2 : n\in \mathbb N \}$$

Not what you are looking for?

You are also free to write something like “$\{a_n : n\in \mathbb N\}$ where $a_1 = 3$ and $a_{n+1} = a_n + 2n + 1$.”

Answer 2

Yes, perfect.

If $\Bbb N$ is used in the context with $0$, then in the set you can use for example $\Bbb N^+$ or $\Bbb Z^+$ or most clearly $\Bbb N\setminus\{0\}$. Or, you can also write $a_0=3$ instead of $a_1=3$ in that case..