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Which of the following sequences are convergent? Select all that apply.

a geometric sequence with r=1/5
an arithmetic sequence with d=-4
a geometric sequence with r = -2
an arithmetic sequence with d=1/5
a geometric sequence with r=2/3

User Linuscl
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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:

  1. A geometric sequence with r=1/5
  2. A geometric sequence with r=2/3
User Luixal
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