Question

4. Given the first five terms of the sequence 19, 15, 11, 7, 3... use the subscript notation to describe a7,
A) a7 = −9
B) a7 = −5
C) a7 = −1
D) a7 = -4


5. Given the first five terms of the sequence 81, 27, 9, 3, 1, ... use the subscript notation to describe a7
A) a7 = 1/27
B) a7 = 27
C) a7 = 1/9
D) a7 = 3

Answer 1

To analyze both sequences and determine the values of \( a_7 \), we first need to identify the pattern in each sequence.

### Problem 4
Given the sequence: 19, 15, 11, 7, 3...

This sequence is a linear sequence where each term decreases by 4. We can express this as:

- \( a_1 = 19 \)
- \( a_2 = 15 = 19 - 4 \)
- \( a_3 = 11 = 19 - 8 \)
- \( a_4 = 7 = 19 - 12 \)
- \( a_5 = 3 = 19 - 16 \)

The general term can be described as:
\[
a_n = 19 - 4(n - 1)
\]
Alternatively, this can be simplified to:
\[
a_n = 23 - 4n
\]

Now, we can find \( a_7 \):
\[
a_7 = 23 - 4(7) = 23 - 28 = -5
\]

Thus, the answer for problem 4 is:
**B) a7 = −5**

### Problem 5
Given the sequence: 81, 27, 9, 3, 1...

This sequence is a geometric sequence where each term is obtained by multiplying the previous term by \(\frac{1}{3}\). We can express this as:

- \( a_1 = 81 \)
- \( a_2 = 27 = 81 \times \frac{1}{3} \)
- \( a_3 = 9 = 27 \times \frac{1}{3} \)
- \( a_4 = 3 = 9 \times \frac{1}{3} \)
- \( a_5 = 1 = 3 \times \frac{1}{3} \)

The general term can be described as:
\[
a_n = 81 \times \left(\frac{1}{3}\right)^{n-1}
\]

Now, we can find \( a_7 \):
\[
a_7 = 81 \times \left(\frac{1}{3}\right)^{7-1} = 81 \times \left(\frac{1}{3}\right)^{6} = 81 \times \frac{1}{729}
\]

Since \( 81 = 3^4 \) and \( 729 = 3^6 \):
\[
a_7 = \frac{3^4}{3^6} = \frac{1}{3^{6-4}} = \frac{1}{3^2} = \frac{1}{9}
\]

Thus, the answer for problem 5 is:
**C) a7 = 1/9**