Final answer:
The convergent sequences from the given options are the geometric sequence with r=1/5 and the geometric sequence with r=2/3, because their common ratios are between -1 and 1.
Step-by-step explanation:
The question is asking us to determine which of the given sequences are convergent sequences. A sequence is convergent if it approaches a specific value as it progresses to infinity. Now, let's analyze each sequence separately:
- A geometric sequence with r=1/5 is convergent because the ratio (r) is between -1 and 1, which means the terms of the sequence will get closer and closer to 0 as the sequence progresses.
- An arithmetic sequence with d=-4 is not convergent because the difference (d) is constant, and the terms will continue to decrease indefinitely without approaching a specific value.
- A geometric sequence with r = -2 is not convergent because the absolute value of the ratio (r) is greater than 1, causing the terms to grow in absolute value indefinitely without approaching a specific value.
- An arithmetic sequence with d=1/5 is not convergent for the same reason as the sequence with d=-4; the terms will continue to increase indefinitely without approaching a specific value.
- A geometric sequence with r=2/3 is convergent, again because the ratio (r) is between -1 and 1, which ensures that the terms decrease in absolute value, approaching 0.
Summing up, the convergent sequences are:
- A geometric sequence with r=1/5
- A geometric sequence with r=2/3