Classify the sequence left-brace a subscript n baseline right-brace equals left-brace 3 comma negative 1 comma Start Fraction 1 over 3 End Fraction comma negative Start Fraction 1 over 9 End Fraction ellipsis right-brace as arithmetic, geometric, or neither. If there is not enough information to classify the sequence, choose not enough information.
To classify the sequence, we need to determine if there is a common difference between consecutive terms or a common ratio between consecutive terms.
The given sequence is: {a_n} = {3, -1, 1/3, -1/9, ...}
To check if it is an arithmetic sequence, we need to see if there is a common difference between consecutive terms. Let's calculate the differences:
-1 - 3 = -4
1/3 - (-1) = 4/3
-1/9 - (1/3) = -4/9
The differences are not constant, so the sequence is not arithmetic.
To check if it is a geometric sequence, we need to see if there is a common ratio between consecutive terms. Let's calculate the ratios:
-1 / 3 = -1/3
1/3 / (-1) = -1/3
-1/9 / (1/3) = -1/3
The ratios are constant, so the sequence is geometric.
Therefore, the given sequence {a_n} = {3, -1, 1/3, -1/9, ...} is a geometric sequence.