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.
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.