Question

Question 1 A)Given the recursive formula for the arithmetic sequence a1=−5, an=an−1+4, find the first three terms of the sequence.(2 points) a1=$$, a2=$$, and a3=$$ Question 2 A) Write out the first five terms of the sequence an=(−2)n. Determine if the sequence is arithmetic or geometric, and then find the common ratio or difference. Which option below is accurate? Option #1: The sequence is arithmetic, and the common difference is 6. Option #2: The sequence is arithmetic, and the common difference is −2. Option #3: The sequence is geometric, and the common ratio is −2. Option #4: The sequence is geometric, and the common ratio is 2. (1 point) Option #$$ is accurate. Question 3 A)What is the common difference in the recursively defined arithmetic sequence a1=4, an=an−1−5.(1 point) The common difference is $$. Question 4 A) Which option is the correct recursive equation for the arithmetic sequence 6, 2, −2, −6, . . .? Option #1: a1=6, an=an−1+4 Option #2: a1=6, an=an−1−4 Option #3: a1=6, an=4an−1 Option #4: a1=6, an=−4an−1 (1 point) Option #$$ is the correct recursive formula for the sequence. Question 5 A)Find the 7th value in the arithmetic sequence described by the explicit formula An=2.5+(n−1)4.(1 point) A7=$$

Answer 1

Let's solve the questions step by step:

### Question 1
**Given:**
- \( a_1 = -5 \)
- \( a_n = a_{n-1} + 4 \)

**Finding the first three terms:**
1. \( a_1 = -5 \)
2. \( a_2 = a_1 + 4 = -5 + 4 = -1 \)
3. \( a_3 = a_2 + 4 = -1 + 4 = 3 \)

Thus,
- \( a_1 = -5 \)
- \( a_2 = -1 \)
- \( a_3 = 3 \)

---

### Question 2
**Sequence:** \( a_n = (-2)^n \)

**Finding the first five terms:**
1. \( a_1 = (-2)^1 = -2 \)
2. \( a_2 = (-2)^2 = 4 \)
3. \( a_3 = (-2)^3 = -8 \)
4. \( a_4 = (-2)^4 = 16 \)
5. \( a_5 = (-2)^5 = -32 \)

The first five terms are: -2, 4, -8, 16, -32.

**Determining if the sequence is arithmetic or geometric:**

- **Arithmetic:** The common difference is calculated as \( a_n - a_{n-1} \).
- \( 4 - (-2) = 6 \)
- \( -8 - 4 = -12 \)
- \( 16 - (-8) = 24 \)
- \( -32 - 16 = -48 \)

The differences are not constant, hence not arithmetic.

- **Geometric:** The common ratio is calculated as \( \frac{a_n}{a_{n-1}} \).
- \( \frac{4}{-2} = -2 \)
- \( \frac{-8}{4} = -2 \)
- \( \frac{16}{-8} = -2 \)
- \( \frac{-32}{16} = -2 \)

The common ratio is constant and equals -2.

Thus, the sequence is geometric, with the common ratio of -2.

**Correct option:**
- Option #3: The sequence is geometric, and the common ratio is −2.

---

### Question 3
**Given:**
- \( a_1 = 4 \)
- \( a_n = a_{n-1} - 5 \)

**Finding the common difference:**
The common difference in an arithmetic sequence defined recursively as \( a_n = a_{n-1} + d \) is simply the value being added, which is -5 in this case.

Thus,
- The common difference is -5.

---

### Question 4
**Finding the correct recursive equation for the sequence 6, 2, −2, −6, . . .:**

The sequence decreases from 6 to 2, then from 2 to -2, and from -2 to -6.

Calculating the differences:
- \( 2 - 6 = -4 \)
- \( -2 - 2 = -4 \)
- \( -6 - (-2) = -4 \)

This shows that the common difference is -4.

**Checking the options:**
- Option #1: \( a_1 = 6, a_n = a_{n-1} + 4 \) (not correct - incorrect difference)
- Option #2: \( a_1 = 6, a_n = a_{n-1} - 4 \) (correct)
- Option #3: \( a_1 = 6, a_n = 4a_{n-1} \) (not correct)
- Option #4: \( a_1 = 6, a_n = -4a_{n-1} \) (not correct)

**Thus, the correct recursive formula is Option #2.**

---

### Question 5
**Explicit formula:** \( A_n = 2.5 + (n-1) \cdot 4 \)

**Finding \( A_7 \):**
\[
A_7 = 2.5 + (7-1) \cdot 4
\]
\[
= 2.5 + 6 \cdot 4
\]
\[
= 2.5 + 24
\]
\[
= 26.5
\]

Thus,
- \( A_7 = 26.5 \)

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### Summary of Answers:
1. \( a_1 = -5, a_2 = -1, a_3 = 3 \)
2. **Option #3** is accurate.
3. The common difference is -5.
4. **Option #2** is the correct recursive formula for the sequence.
5. \( A_7 = 26.5 \)