Question

Which of the following sequences are convergent? Select all that apply.
(1 point)
Responses

a geometric sequence with r equals Start Fraction 1 over 5 End Fraction
a geometric sequence with Image with alt text: r equals Start Fraction 1 over 5 End Fraction

an arithmetic sequence with d equals negative 4
an arithmetic sequence with Image with alt text: d equals negative 4

a geometric sequence with r equals negative 2
a geometric sequence with Image with alt text: r equals negative 2

an arithmetic sequence with d equals Start Fraction 1 over 5 End Fraction
an arithmetic sequence with Image with alt text: d equals Start Fraction 1 over 5 End Fraction

a geometric sequence with r equals Start Fraction 2 over 3 End Fraction
a geometric sequence with Image with alt text: r equals Start Fraction 2 over 3 End Fraction

Answer 1

To determine the convergence of the sequences listed, we need to analyze both geometric and arithmetic sequences.

1. **Geometric Sequence:**
- A geometric sequence converges if the absolute value of the common ratio \( r \) is less than 1 (\(|r| < 1\)).
- If \( |r| \geq 1 \), the geometric sequence diverges.

2. **Arithmetic Sequence:**
- An arithmetic sequence diverges because it continues to increase or decrease without bound (the terms do not approach a fixed value).

### Analysis of Given Sequences:

1. **a geometric sequence with \( r = \frac{1}{5} \)**:
- \( |r| = \frac{1}{5} < 1 \)
- **Convergent**

2. **an arithmetic sequence with \( d = -4 \)**:
- This is an arithmetic sequence and it diverges.
- **Divergent**

3. **a geometric sequence with \( r = -2 \)**:
- \( |r| = 2 \geq 1 \)
- **Divergent**

4. **an arithmetic sequence with \( d = \frac{1}{5} \)**:
- This is an arithmetic sequence and it diverges.
- **Divergent**

5. **a geometric sequence with \( r = \frac{2}{3} \)**:
- \( |r| = \frac{2}{3} < 1 \)
- **Convergent**

### Summary of Convergence:

- **Convergent:**
- a geometric sequence with \( r = \frac{1}{5} \)
- a geometric sequence with \( r = \frac{2}{3} \)

- **Divergent:**
- an arithmetic sequence with \( d = -4 \)
- an arithmetic sequence with \( d = \frac{1}{5} \)
- a geometric sequence with \( r = -2 \)

Thus, the sequences that are convergent are:
- a geometric sequence with \( r = \frac{1}{5} \)
- a geometric sequence with \( r = \frac{2}{3} \)