Triangular numbers, proving numbers are tringular

[schlynn]

I've been learning about polygonal numbers, and one of the exercises in this book ask me to show that 9t$_{n}$+1 [Fermat], 25t$_{n}$+3, and 49t$_{n}$+6 [both from Euler] are triangular numbers. I don't know how to approach these proofs, I've tried to show that they have some form similar to $\frac{n(n+1)}{2}$, but with no avail. But it looks like there is a pattern, that would be
(2n+1)$^{2}$t$_{\alpha}$+$\frac{n(n+1)}{2}$, but I have no way of proving this. Could someone point me in the correct direction?

t$_{n}$ and t$_{\alpha}$ are both triangular numbers.

 

[gopher_p]

$9\cdot\frac{n(n+1)}{2}+1=\frac{(3n+1)(3n+2)}{2}=\frac{(3n+1)\big((3n+1)+1 \big)}{2}$, yes?

 

[schlynn]

Wow, I was over thinking this a lot. Thank you, I can check my pattern with this too.

 

[gopher_p]

Another one is $5929t_n+741$ is triangular whenever $t_n$ is triangular [gopher_p].

But it looks like you've already figured out that pattern.