To find the 20th term of the series $-3, -2 \frac{1}{2}, -2, \dots$, let's first determine the pattern of the series.

The series is an arithmetic sequence where the first term ($a_1$) is $-3$ and the common difference ($d$) can be found by subtracting the first term from the second term:

$d = -2 \frac{1}{2} - (-3) = -2.5 + 3 = 0.5$

Thus, the common difference $d$ is $0.5$.

The formula for the $n$th term of an arithmetic sequence is:

$a_n = a_1 + (n-1) \cdot d$

Substituting the given values:

$a_{20} = -3 + (20-1) \cdot 0.5$

Simplifying:

$a_{20} = -3 + 19 \cdot 0.5 = -3 + 9.5 = 6.5$

So, the 20th term of the series is $6.5$.

Would you like more details or have any questions?

Here are some related questions:

1. What is the formula for the sum of the first $n$ terms of an arithmetic series?
2. How would you find the 50th term in this series?
3. Can you determine the common difference if you only know the first and last term of an arithmetic series?
4. How does the series change if the common difference is negative?
5. What if the sequence is geometric instead of arithmetic?

Tip: When identifying the pattern in a sequence, it's important to check if the differences between terms are consistent. This consistency indicates an arithmetic sequence.