Question
Classify the sequence left-brace a subscript n baseline right-brace equals left-brace 4 comma Start Fraction 10 over 3 End Fraction comma Start Fraction 8 over 3 End Fraction comma 2 comma ellipsis right-brace as arithmetic or geometric. Then, determine whether the sequence is convergent or divergent.
Answers
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GPT 3.5
The given sequence is {a_n} = {4, 10/3, 8/3, 2, ...}.
To determine if the sequence is arithmetic or geometric, we need to check if the differences between consecutive terms are constant.
First, let's find the differences between consecutive terms:
d_1 = 10/3 - 4 = 2/3
d_2 = 8/3 - 10/3 = -2/3
d_3 = 2 - 8/3 = 6/3 - 8/3 = -2/3
Since the differences between consecutive terms are constant and equal to -2/3, we can conclude that the sequence is arithmetic.
Next, we need to check if the sequence is convergent or divergent.
In an arithmetic sequence, if the common difference (d) is non-zero, the sequence is divergent.
In this case, the common difference (d) is -2/3, which is non-zero. Therefore, the sequence is divergent.
To determine if the sequence is arithmetic or geometric, we need to check if the differences between consecutive terms are constant.
First, let's find the differences between consecutive terms:
d_1 = 10/3 - 4 = 2/3
d_2 = 8/3 - 10/3 = -2/3
d_3 = 2 - 8/3 = 6/3 - 8/3 = -2/3
Since the differences between consecutive terms are constant and equal to -2/3, we can conclude that the sequence is arithmetic.
Next, we need to check if the sequence is convergent or divergent.
In an arithmetic sequence, if the common difference (d) is non-zero, the sequence is divergent.
In this case, the common difference (d) is -2/3, which is non-zero. Therefore, the sequence is divergent.