Question

Question 1 A)Use the explicit formula an=4−3nan=4−3n to find the seventh term of the arithmetic sequence.(1 point) Responses a7=1a7=1 a subscript 7 baseline equals 1 a7=7a7=7 a subscript 7 baseline equals 7 a7=−17a7=−17 a subscript 7 baseline equals negative 17 a7=−1a7=−1 a subscript 7 baseline equals negative 1 Question 2 A)Given the recursive formula for the geometric sequence a1=5a1=5, an=25an−1an=25an−1, find the second term of the sequence.(1 point) Responses a2=2a2=2 a subscript 2 baseline equals 2 a2=525a2=525 a subscript 2 baseline equals 5 Start Fraction 2 over 5 End Fraction a2=25a2=25 a subscript 2 baseline equals Start Fraction 2 over 5 End Fraction a2=85a2=85 a subscript 2 baseline equals Start Fraction 8 over 5 End Fraction Question 3 A)Use the recursively defined geometric sequence a1=56a1=56, an=4an−1an=4an−1 and find the common ratio.(1 point) Responses 4 4 5656 Start Fraction 5 over 6 End Fraction −23−23 negative Start Fraction 2 over 3 End Fraction 103103 Start Fraction 10 over 3 End Fraction Question 4 A)Find the recursive formula for the arithmetic sequence 7, 5, 3, 1, −1−1, . . .(1 point) Responses a1=7a1=7, an=−2an−1an=−2an−1 a subscript 1 baseline equals 7 , a subscript n baseline equals negative 2 a subscript n minus 1 baseline a1=7a1=7, an=an−1−2an=an−1−2 a subscript 1 baseline equals 7 , a subscript n baseline equals a subscript n minus 1 baseline minus 2 a1=7a1=7, an=2an−1an=2an−1 a subscript 1 baseline equals 7 , a subscript n baseline equals 2 a subscript n minus 1 baseline a1=7a1=7, an=an−1+2an=an−1+2 a subscript 1 baseline equals 7 , a subscript n baseline equals a subscript n minus 1 baseline plus 2 Question 5 A)Which of the following correctly uses the explicit formula, An=14+(n−1)⋅12An=14+(n−1)⋅12, to find the 9th term of the described arithmetic sequence?(1 point) Responses 434434 4 Start Fraction 3 over 4 End Fraction 414414 4 Start Fraction 1 over 4 End Fraction 814814 8 Start Fraction 1 over 4 End Fraction 834834 8 Start Fraction 3 over 4 End Fraction

Answer 1

5 question: explicit formula, An= 1/4 + (n-1) * 1/2
2 question: a1=5, an=2/5an-1

Answer 2

question 5 responses are
4 3/4
4 1/4
8 1/4
8 3/4

Answer 3

Let's solve each question step by step.

### Question 1:
**Given**: \( a_n = 4 - 3n \)

To find the 7th term, we calculate \( a_7 \):
\[
a_7 = 4 - 3(7) = 4 - 21 = -17
\]
**Answer**: \( a_7 = -17 \)

### Question 2:
**Given**: \( a_1 = 5 \) and \( a_n = 25a_{n-1} \)

To find the second term \( a_2 \):
\[
a_2 = 25a_1 = 25 \times 5 = 125
\]
**Answer**: \( a_2 = 125 \)

### Question 3:
**Given**: \( a_1 = 56 \) and \( a_n = 4a_{n-1} \)

Here, the common ratio \( r \) is simply the multiplier we use to go from one term to the next:
\[
r = 4
\]
**Answer**: \( r = 4 \)

### Question 4:
The sequence is: 7, 5, 3, 1, -1...

To determine the recursive formula, we notice that the difference between consecutive terms is -2:
- From 7 to 5: \(-2\)
- From 5 to 3: \(-2\)
- From 3 to 1: \(-2\)
- From 1 to -1: \(-2\)

Thus, the recursive formula is:
\[
a_1 = 7, \quad a_n = a_{n-1} - 2
\]
**Answer**: \( a_1 = 7, a_n = a_{n-1} - 2 \)

### Question 5:
**Given**: \( A_n = 14 + (n - 1) \cdot 12 \)

To find the 9th term:
\[
A_9 = 14 + (9 - 1) \cdot 12 = 14 + (8) \cdot 12 = 14 + 96 = 110
\]
**Answer**: \( A_9 = 110 \)

As a recap the answers to each respective question are:
1. \( a_7 = -17 \)
2. \( a_2 = 125 \)
3. Common ratio \( r = 4 \)
4. \( a_1 = 7, a_n = a_{n-1} - 2 \)
5. \( A_9 = 110 \)

Answer 4

Thank you for the clarification. Let's solve the specified questions using the provided formulas.

### Question 5 (Explicit Formula):
**Given**: \( A_n = \frac{1}{4} + (n - 1) \cdot \frac{1}{2} \)

To find the 9th term \( A_9 \):
\[
A_9 = \frac{1}{4} + (9 - 1) \cdot \frac{1}{2}
\]

Calculating it step by step:
\[
A_9 = \frac{1}{4} + 8 \cdot \frac{1}{2}
\]
\[
A_9 = \frac{1}{4} + 4
\]
Converting 4 to fractional form:
\[
A_9 = \frac{1}{4} + \frac{16}{4} = \frac{1 + 16}{4} = \frac{17}{4}
\]

Thus, the answer for Question 5 is:
**Answer**: \( A_9 = \frac{17}{4} \)

---

### Question 2 (Recursive Formula):
**Given**: \( a_1 = 5 \) and \( a_n = \frac{2}{5} a_{n-1} \)

To find the second term \( a_2 \):
\[
a_2 = \frac{2}{5} a_1 = \frac{2}{5} \cdot 5 = 2
\]

Thus, the answer for Question 2 is:
**Answer**: \( a_2 = 2 \)

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### Summary of Answers:
1. **Question 5**: \( A_9 = \frac{17}{4} \)
2. **Question 2**: \( a_2 = 2 \)

If you have any more questions or need further clarification, feel free to ask!

Answer 5

Thank you for providing the response options for Question 5. Let's correctly determine the 9th term using the explicit formula given and match it with the provided choices.

### Question 5:
**Given explicit formula**:
\[
A_n = \frac{1}{4} + (n - 1) \cdot \frac{1}{2}
\]

To find the 9th term \( A_9 \):
\[
A_9 = \frac{1}{4} + (9 - 1) \cdot \frac{1}{2}
\]
\[
= \frac{1}{4} + 8 \cdot \frac{1}{2}
\]
\[
= \frac{1}{4} + 4
\]
Convert 4 into fourths for easier addition:
\[
4 = \frac{16}{4}
\]
So,
\[
A_9 = \frac{1}{4} + \frac{16}{4} = \frac{17}{4}
\]

Now, let's convert \(\frac{17}{4}\) into a mixed number to compare:
\[
\frac{17}{4} = 4 \frac{1}{4}
\]

Thus, among the response options, the correct term is:
**Answer**: \( 4 \frac{1}{4} \)

Thank you for your patience! Let me know if you need any further assistance.