Determine whether each sequence below is arithmetic, geometric, or neither. Provide support for your conclusions.
Sequence 1: 1/2, 7/6, 11/6, 5/2, ...
Sequence 2: 1/2, 1/3, 2/9, 4/27, ...
To determine whether each sequence is arithmetic, geometric, or neither, we need to identify the pattern of the differences or ratios between consecutive terms.
Sequence 1: 1/2, 7/6, 11/6, 5/2, ...
Let's calculate the differences between each consecutive pair of terms:
7/6 - 1/2 = (7 - 3) / 6 = 4/6 = 2/3
11/6 - 7/6 = (11 - 7) / 6 = 4/6 = 2/3
5/2 - 11/6 = (15 - 11) / 6 = 4/6 = 2/3
The differences between consecutive terms are all the same, 2/3. Therefore, this sequence is arithmetic.
Sequence 2: 1/2, 1/3, 2/9, 4/27, ...
Let's calculate the ratios between each consecutive pair of terms:
(1/3) / (1/2) = (1/3) * (2/1) = 2/3
(2/9) / (1/3) = (2/9) * (3/1) = 2/3
(4/27) / (2/9) = (4/27) * (9/2) = 2/3
The ratios between consecutive terms are all the same, 2/3. Therefore, this sequence is geometric.
In conclusion:
- Sequence 1 is arithmetic because the differences between consecutive terms are the same.
- Sequence 2 is geometric because the ratios between consecutive terms are the same.