To find the general term for the arithmetic sequence, we use the formula for the $n$-th term of an arithmetic sequence:

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

Given:

• $a_1 = 1$ (the first term)
• $d = -8$ (the common difference)

Step 1: Substitute the values into the formula

$a_n = 1 + (n - 1) \cdot (-8)$

Step 2: Simplify the expression

$a_n = 1 - 8(n - 1)$ $a_n = 1 - 8n + 8$ $a_n = -8n + 9$

So, the general formula for the $n$-th term is:

$a_n = -8n + 9$

Step 3: Find the 20th term ($a_{20}$)

Now, substitute $n = 20$ into the formula:

$a_{20} = -8(20) + 9$ $a_{20} = -160 + 9$ $a_{20} = -151$

Summary

• The general formula for the $n$-th term is: $a_n = -8n + 9$
• The 20th term, $a_{20}$, is: $a_{20} = -151$

Would you like further explanation on any step or have additional questions?

Here are some related questions to deepen understanding:

1. How would the formula change if the common difference were positive?
2. What is the difference between an arithmetic sequence and a geometric sequence?
3. How can we find the sum of the first 20 terms in this sequence?
4. How would you find the term when $n = 50$ in this sequence?
5. How does the sign of the common difference affect the progression of terms in the sequence?

Tip: In arithmetic sequences, each term increases or decreases by the common difference $d$. The direction of change (positive or negative) depends on whether $d$ is positive or negative.