$$1 + 3 + 5 + \dots + (2n + 1) $$
For the above question, the answer is $(n + 1)^2$ and I understand that $n$ is the number of terms. If I let my $n$ is $3$, that means I add $1 + 3 + 5 = 9$ but if I add inside the equation I get $$(3 + 1)^2= 16$$ which is wrong.
Did I misunderstanding the answer? Can someone advise?
$n$ is not the number of terms; $2n+1$ is the value of the last term. So for your example,
$$1+3+5 = 1 + 3 + \cdots + (2\cdot\color{red}2+1) = (\color{red}2+1)^2 = 9$$
And as you can see, when $n=0$, there is one term; when $n = 2$, there are three terms; so the number of terms is $n+1$.
Also, compare $2n+1$ with the general formula for the $k$th term of an arithmetic sequence:
$$a + r(k-1)$$ where $a$ is the first term and $r$ is the term difference.