Let's break down and solve the problems based on the image you've uploaded.

Given Sequence:

$-8, -11, -14, -17, -20, \ldots$

1. Common Difference (d):

The common difference $d$ in an arithmetic sequence is the difference between any two consecutive terms.

For example: $-11 - (-8) = -11 + 8 = -3$ $-14 - (-11) = -14 + 11 = -3$ So, the common difference $d = -3$.

2. Simplified Formula for the Arithmetic Sequence:

The formula for the $n$-th term of an arithmetic sequence is: $f(n) = a_1 + (n - 1) \cdot d$ Where:

• $a_1$ is the first term of the sequence,
• $d$ is the common difference,
• $n$ is the term number.

In this case:

• $a_1 = -8$,
• $d = -3$.

Thus, the formula becomes: $f(n) = -8 + (n - 1) \cdot (-3)$ Simplifying: $f(n) = -8 - 3(n - 1) = -8 - 3n + 3 = -3n - 5$

So, the simplified formula is: $f(n) = -3n - 5$

3. Find the 30th Term $f(30)$:

Using the formula $f(n) = -3n - 5$, we substitute $n = 30$: $f(30) = -3(30) - 5 = -90 - 5 = -95$

Thus, the 30th term is $f(30) = -95$.

Let me know if you'd like further details or have any questions!

Here are 5 related questions:

1. How do you find the sum of the first 30 terms of this arithmetic sequence?
2. What is the 50th term of the sequence using the same formula?
3. How can you determine the general term of any arithmetic sequence?
4. What happens to the terms if the common difference is positive?
5. How would the sequence look if you started with a different first term?

Tip: For arithmetic sequences, always check consecutive terms to ensure the difference is constant!