The following arithmetic sequence is given.
 
14, 11, 8, 5, ...

What are the next three terms?
A.4, 3, 2

B.2, −1, −4

C.3, 1, −2

D.1, −1, −3

1b. 
Which of the following is the correct equation for the nth term of the following arithmetic sequence?
A.an = 3n + 17

B.an = −3n + 17

C.an = 3n − 17

D.an = 17n − 3


To find the next three terms of the arithmetic sequence \(14, 11, 8, 5, \ldots\), we first determine the common difference:

\[
\text{Common difference} = 11 - 14 = -3
\]

So, each term decreases by 3. Continuing this pattern:

1. \(5 - 3 = 2\)
2. \(2 - 3 = -1\)
3. \(-1 - 3 = -4\)

Thus, the next three terms are \(2, -1, -4\). The correct answer is **B. 2, −1, −4**.

### For part 1b:
To find the equation for the \(n\)th term of the sequence, we can use the formula for the \(n\)th term of an arithmetic sequence, which is given by:

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

Where:
- \(a_1 = 14\) (the first term),
- \(d = -3\) (the common difference).

Substituting the values, we get:

\[
a_n = 14 + (n - 1)(-3)
\]

Simplifying this:

\[
a_n = 14 - 3(n - 1) = 14 - 3n + 3 = 17 - 3n
\]

Rearranging gives:

\[
a_n = -3n + 17
\]

So, the correct equation for the \(n\)th term is **B. \(a_n = -3n + 17\)**.

### Summary:
1. The next three terms are \(2, -1, -4\) (B).
2. The equation for the \(n\)th term is \(a_n = -3n + 17\) (B).

Would you like any further details or have any questions? Here are some follow-up questions for you:

1. How do you derive the common difference in an arithmetic sequence?
2. Can you explain how to find the first term and common difference from a given sequence?
3. What are some applications of arithmetic sequences in real life?
4. How would the sequence change if the common difference was positive instead?
5. What happens if the sequence is not linear? How would you determine the pattern?

**Tip:** When analyzing sequences, always look for a consistent pattern in the differences between consecutive terms.

The following arithmetic sequence is given: 14, 11, 8, 5, ... What are the next three terms? Which of the following is the correct equation for the nth term of the following arithmetic sequence?

Explore how to find the next three terms of the arithmetic sequence 14, 11, 8, 5 and derive the nth term equation. This solution covers key concepts such as common differences and the formula for arithmetic sequences. Ideal for students in Grades 6-8.

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