In an arithmetic sequence, the general term (or $n$-th term) is given by:

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

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $d$ is the common difference.

We are given:

• The 3rd term ($a_3 = -12$),
• The 7th term ($a_7 = 8$).

Step 1: Write the expressions for the 3rd and 7th terms

For the 3rd term: $a_3 = a_1 + 2d = -12$ For the 7th term: $a_7 = a_1 + 6d = 8$

Step 2: Solve the system of equations

We now have two equations:

1. $a_1 + 2d = -12$
2. $a_1 + 6d = 8$

Subtract the first equation from the second to eliminate $a_1$:

$(a_1 + 6d) - (a_1 + 2d) = 8 - (-12)$

This simplifies to: $4d = 20$

So, the common difference is: $d = 5$

Step 3: Find $a_1$

Substitute $d = 5$ into the first equation: $a_1 + 2(5) = -12$ $a_1 + 10 = -12$ $a_1 = -22$

Step 4: Find the 20th term

Now, use the general formula to find the 20th term: $a_{20} = a_1 + (20-1) d = -22 + 19(5)$ $a_{20} = -22 + 95 = 73$

Thus, the 20th term is: $\boxed{73}$

Would you like further explanation or details on any step?

Here are 5 related questions to consider:

1. How can you verify if a given sequence is arithmetic?
2. What happens if the common difference is negative in an arithmetic sequence?
3. How do you derive the sum of the first $n$ terms of an arithmetic sequence?
4. How would the sequence change if the 7th term was different, say 10 instead of 8?
5. Can you derive the first term of an arithmetic sequence from just two known terms?

Tip: To quickly check if a sequence is arithmetic, subtract consecutive terms and see if the difference is constant.