TITLE: The cofinality of $\aleph_{\omega\cdot9+3}$
QUESTION [5 upvotes]: I am studying for a test and I was able to find the cofinality 3 of the 4 ones given, but am having a lot of trouble with the 4th.
the 3 first ones are:

$\newcommand{\cf}{\operatorname{cf}}\cf(\aleph_\omega)=\omega$
$\cf(\aleph_{\omega^2})=\omega$
$\cf(\omega_1)=\omega_1$

I do not know how to calcuate $\cf(\aleph_{\omega\cdot9 + 3})$. I am very tempted to say it is simply $\omega$.
Thank you.

REPLY [3 votes]: Like you, I initially thought $\omega$ is enough. So I tried the sequence $\aleph_{n\cdot 9+3}$, it turns out all of them are strictly smaller than $\aleph_{\omega}$, which violates the conditions of a cofinality.
I won't spoil the solution to the general case of this problem, because Asaf Karagila has provided copious hints.